Solvability of superlinear fractional parabolic equations

Abstract

We study necessary conditions and sufficient conditions for the existence of local-in-time solutions of the Cauchy problem for superlinear fractional parabolic equations. Our conditions are sharp and clarify the relationship between the solvability of the Cauchy problem and the strength of the singularities of the initial measure.

1 Introduction

We consider the Cauchy problem for a superlinear fractional parabolic equation

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u+(-\Delta )^{\frac{\theta }{2}} u=F(u), &{}\quad x\in {{ \mathbb {R}}}^N,\; \, t>0,\\ u(0)=\mu &{} \quad \hbox {in} \quad {{ \mathbb {R}}}^N, \end{array} \right. \end{aligned}$$

(P)

where $\mu$ is a nonnegative Radon measure in ${{ \mathbb {R}}}^N$. Throughout the paper, we assume that $N\ge 1$, $0<\theta \le 2$, and $F:[0,\infty )\rightarrow [0,\infty )$ is (at least) continuous.

In general, the existence of local-in-time nonnegative solutions of problem (P) depends crucially on the delicate interplay between the strength of the singularities of the initial measure $\mu$ and the behavior of $F(\tau )$ as $\tau \rightarrow \infty$. In this paper, for a large class of nonlinearities F, we obtain new necessary conditions and new sufficient conditions for the local solvability of problem (P). The prototypical example we have in mind is

$$\begin{aligned} F(\tau )=\tau ^p[\log (L+\tau )]^q, \hbox { where } p>1, q\in {{ \mathbb {R}}}, \hbox { and }L\ge 1. \end{aligned}$$

As a consequence of our more general results, we are then able to derive sharp results for classes of nonlinearities which include these prototypes as special cases, and quantify this interplay more precisely via ‘optimal singularities.’

Throughout this paper, we use the following notations. For $T>0$, we set $Q_T:={{\mathbb {R}}}^N\times (0,T)$ and let $B(x,\sigma )$ denote the Euclidean ball in ${{ \mathbb {R}}}^N$ center x, radius $\sigma$. We use for the average value of f over B with respect to the Lebesgue measure dx. The set of nonnegative Lebesgue measurable functions in ${\mathbb R}^N$ is denoted by ${{\mathcal {L}}}_0$, while ${{\mathcal {M}}}$ denotes the set of nonnegative Radon measures in ${\mathbb R}^N$. For $\mu \in {\mathcal L}_0$, we abuse terminology somewhat by speaking of ‘measure $\mu$’ defined via ${\text {d}}\mu =\mu (x){\text {d}}x$.

1.1 Background

The solvability of the Cauchy problem for superlinear parabolic equations has been studied in many papers since the pioneering work by Fujita [14]. The literature is now very extensive, and we refer to the comprehensive monograph [35]. We also mention the following works, some of which are directly related to this paper, others with a different emphasis (higher-order equations, systems, nonlinear boundary conditions): superlinear parabolic equations [2, 6, 7, 14, 29,30,31, 33, 36, 38,39,40,41]; linear heat equation with nonlinear boundary conditions [10, 15, 20, 27, 28]; superlinear parabolic equations with a potential [1, 3, 9, 22, 23, 39]; superlinear parabolic systems [11,12,13, 26, 34]; superlinear fractional parabolic equations [18, 19, 21, 32, 37]; superlinear higher-order parabolic equations [8, 16, 17, 24, 25].

In [19], the second and third authors of this paper considered problem (P) in the special case of the power law nonlinearity $F(u)=u^p$ with $p>1$:

$$\begin{aligned} \left\{ \begin{array}{ll} \partial _t u+(-\Delta )^{\frac{\theta }{2}} u=u^p, &{}\quad x\in {{ \mathbb {R}}}^N,\,\,\,t>0,\\ u(0)=\mu &{}\quad \hbox {in} \quad {{ \mathbb {R}}}^N. \end{array} \right. \end{aligned}$$

(1.1)

There, as here, the exponent $p_\theta :=1+\theta /N$ plays a critical role. They proved the following necessary conditions for the local existence (cases (i) and (ii)).

• (i) Let $\mu \in {{\mathcal {M}}}$. If problem (1.1) possesses a nonnegative solution in $Q_T$ for some $T>0$, then there exists $C_1=C_1(N,\theta ,p)>0$ such that

  $$\begin{aligned} \sup _{x\in {{ \mathbb {R}}}^N}\mu (B(x,\sigma ))\le C_1\sigma ^{N-\frac{\theta }{p-1}}, \quad 0<\sigma \le T^{\frac{1}{\theta }}. \end{aligned}$$

  (1.2)

  In the case where $1<p<p_\theta$, the function $(0,\infty )\ni \sigma \mapsto \sigma ^{N-\theta /(p-1)}$ is decreasing so that relation (1.2) is equivalent to

  $$\begin{aligned} \sup _{x\in {{ \mathbb {R}}}^N}\mu \big (B\big (x,T^{\frac{1}{\theta }}\big )\big )\le C_1 T^{\frac{N}{\theta }-\frac{1}{p-1}}. \end{aligned}$$

  In the case where $p=p_\theta$, there exists $C_2=C_2(N,\theta )>0$ such that

  $$\begin{aligned} \sup _{x\in {{ \mathbb {R}}}^N}\mu (B(x,\sigma )) \le C_2\left[ \log \biggr (e+\frac{T^{\frac{1}{\theta }}}{\sigma }\biggr )\right] ^{-\frac{N}{\theta }}, \quad 0<\sigma \le T^{\frac{1}{\theta }}. \end{aligned}$$

  (See [2] for the Laplacian case $\theta =2$.)

Condition (i) implies the following nonexistence result.

• (ii) Let $p\ge p_\theta$. There exists $\gamma =\gamma (N,\theta ,p)>0$ such that if $\mu \in {{\mathcal {L}}}_0$ satisfies

  $$\begin{aligned} \begin{array}{ll} \mu (x)\ge \gamma |x|^{-N}\displaystyle {\biggr [\log \left( e+\frac{1}{|x|}\right) \biggr ]^{-\frac{N}{\theta }-1}} &{}\quad \hbox {if}\quad \displaystyle {p=p_\theta },\\ \mu (x)\ge \gamma |x|^{-\frac{\theta }{p-1}} &{}\quad \hbox {if}\quad \displaystyle {p>p_\theta }, \end{array} \end{aligned}$$

  for almost all (a.a.) x in a neighborhood of the origin, then problem (1.1) possesses no local-in-time nonnegative solutions.

Regarding sufficiency, in [19] they obtained results (iii) and (iv) below.

• (iii) Let $\mu \in {{\mathcal {M}}}$ and $1<p<p_\theta$. There exists $c=c(N,\theta ,p)>0$ such that if

  $$\begin{aligned} \sup _{x\in {{ \mathbb {R}}}^N}\mu \big (B\big (x,T^{\frac{1}{\theta }}\big )\big )\le c T^{\frac{N}{\theta }-\frac{1}{p-1}} \end{aligned}$$

  for some $T>0$, then problem (1.1) possesses a nonnegative solution in $Q_T$.

• (iv) Let $\mu \in {{\mathcal {L}}}_0$ and $p\ge p_\theta$. There exists $\varepsilon =\varepsilon (N,\theta ,p) >0$ such that if

  $$\begin{aligned} \begin{array}{ll} 0\le \mu (x)\le \varepsilon |x|^{-N}\displaystyle {\biggr [\log \left( e+\frac{1}{|x|}\right) \biggr ]^{-\frac{N}{\theta }-1}}+K &{}\quad \hbox {if}\quad \displaystyle {p=p_\theta },\\ 0\le \mu (x)\le \varepsilon |x|^{-\frac{\theta }{p-1}}+K &{}\quad \hbox {if}\quad \displaystyle {p>p_\theta }, \end{array} \end{aligned}$$

  for a.a. $x\in {{\mathbb {R}}}^N$ for some $K>0$, then problem (1.1) possesses a local-in-time nonnegative solution.

For $\mu \in {{\mathcal {L}}}_0$, the results in (ii) and (iv) demonstrate that the ‘strength’ of the singularity at the origin of the functions

$$\begin{aligned} \mu _c(x)= \left\{ \begin{array}{ll} |x|^{-\frac{\theta }{p-1}} &{} \quad \hbox {if}\quad p>p_\theta ,\\ |x|^{-N}|\log |x||^{-\frac{N}{\theta }-1} &{} \quad \hbox {if}\quad p=p_\theta , \end{array} \right. \end{aligned}$$

is the critical threshold for the local solvability of problem (1.1). We term such a singularity in the initial data an optimal singularity for the solvability for problem (1.1). Of course, by translation invariance the singularity could be located at any point of ${{ \mathbb {R}}}^N$.

Subsequently, the results of [19] were extended to some related parabolic problems with a power law nonlinearity (see [20,21,22,23,24,25]). However, one cannot apply the arguments in these papers to problem (P) with a general nonlinearity F since they depend heavily upon the homogeneous structure of the power law nonlinearity.

1.2 The main result

In this paper, we improve the arguments in [19] to obtain necessary conditions and sufficient conditions for the existence of local-in-time solutions of problem (P) for a significantly larger class of nonlinearities F and determine the optimal singularities of the initial data for the solvability of problem (P).

Let $f_1$ and $f_2$ be real-valued functions defined in an interval $[L,\infty )$, where $L\in {{ \mathbb {R}}}$. We write $f_1(t)\preceq f_2(t)$ as $t\rightarrow \infty$ if there exists $C>0$ such that $f_1(t)\le Cf_2(t)$ for all large enough $t\in [L,\infty )$. We define $\succeq$ in the obvious way, namely $f_2(t)\succeq f_1(t)$ as $t\rightarrow \infty$ if and only if $f_1(t)\preceq f_2(t)$ as $t\rightarrow \infty$. We write $f_1(t)\asymp f_2(t)$ as $t\rightarrow \infty$ whenever $f_1(t)\preceq f_2(t)$ and $f_1(t)\succeq f_2(t)$ as $t\rightarrow \infty$, i.e., there exists $C>0$ such that $C^{-1}f_2(t)\le f_1(t)\le Cf_2(t)$ for large enough $t\in [L,\infty )$.

We consider nonlinearities which are asymptotic to the prototypical example (F), in this sense:

• (F1) F is locally Lipschitz continuous in $[0,\infty )$;

• (F2) $F(\tau )\asymp \tau ^p[\log \tau ]^q$ as $\tau \rightarrow \infty$ for some $p>1$ and $q\in {{ \mathbb {R}}}$.

Theorem 1.1

Assume conditions (F1) and (F2).

• (i) Let $\mu \in {{\mathcal {M}}}$ and either

  $$\begin{aligned} {\mathrm{(i)}}\quad 1<p<p_\theta \qquad \hbox {or}\qquad {\mathrm{(ii)}}\quad p=p_\theta \hbox { and } q<-1 . \end{aligned}$$

  Problem (P) possesses a local-in-time solution if and only if ${\sup _{z\in {{ \mathbb {R}}}^N}\mu (B(z,1))<\infty }$.

• (ii) Suppose $\mu \in {{\mathcal {L}}}_0$.

  • (1) Let $p=p_\theta$ and $q=-1$. There exists $\gamma _1>0$ such that if

    $$\begin{aligned} \mu (x)\ge \gamma _1|x|^{-N}|\log |x||^{-1}[\log |\log |x||]^{-\frac{N}{\theta }-1} \end{aligned}$$

    (1.3)

    in a neighborhood of $x=0$, then problem (P) possesses no local-in-time solutions. On the other hand, for any $R\in (0,1)$, there exists $\varepsilon _1>0$ such that if

    $$\begin{aligned} 0\le \mu (x)\le \varepsilon _1|x|^{-N}|\log |x||^{-1}[\log |\log |x||]^{-\frac{N}{\theta }-1} \chi _{B(0,R)}(x)+K_1, \quad x\in {{\mathbb {R}}}^N,\nonumber \\ \end{aligned}$$

    (1.4)

    for some $K_1>0$, then problem (P) possesses a local-in-time solution.

  • (2) Let $p=p_\theta$ and $q>-1$. There exists $\gamma _2>0$ such that if

    $$\begin{aligned} \mu (x)\ge \gamma _2|x|^{-N}|\log |x||^{-\frac{N(q+1)}{\theta }-1} \end{aligned}$$

    (1.5)

    in a neighborhood of $x=0$, then problem (P) possesses no local-in-time solutions. On the other hand, for any $R\in (0,1)$, there exists $\varepsilon _2>0$ such that if

    $$\begin{aligned} 0\le \mu (x)\le \varepsilon _2|x|^{-N}|\log |x||^{-\frac{N(q+1)}{\theta }-1}\chi _{B(0,R)}(x)+K_2, \quad x\in {{ \mathbb {R}}}^N, \end{aligned}$$

    (1.6)

    for some $K_2>0$, then problem (P) possesses a local-in-time solution.

  • (3) Let $p>p_\theta$. There exists $\gamma _3>0$ such that if

    $$\begin{aligned} \mu (x)\ge \gamma _3|x|^{-\frac{\theta }{p-1}}|\log |x||^{-\frac{q}{p-1}} \end{aligned}$$

    (1.7)

    in a neighborhood of $x=0$, then problem (P) possesses no local-in-time solutions. On the other hand, for any $R\in (0,1)$, there exists $\varepsilon _3>0$ such that if

    $$\begin{aligned} 0\le \mu (x)\le \varepsilon _3|x|^{-\frac{\theta }{p-1}}|\log |x||^{-\frac{q}{p-1}}\chi _{B(0,R)}(x)+K_3, \quad x\in {{ \mathbb {R}}}^N, \end{aligned}$$

    (1.8)

    for some $K_3>0$, then problem (P) possesses a local-in-time solution.

While Theorem 1.1 provides sharp results on the identification of optimal singularities for the solvability of problem (P), we point out that we have obtained several other interesting and powerful results in this paper regarding necessary conditions and sufficient conditions for existence under very general conditions on F. We mention, in particular, Theorems 3.1, 4.1, 4.2, and 4.3.

Subject to mild assumptions on F (essentially that of majorizing a convex function with suitable monotonicity properties), we follow the strategy in [19] and obtain necessary conditions for the existence in Theorem 3.1. However, the iteration step in [19] to obtain the estimate for the optimal singularity relies on the homogeneity of the pure power law nonlinearity considered there. For the class of nonlinearities satisfying (F1)–(F2), we combine the arguments in [19] with the method introduced in [32], to obtain a sharper necessary condition in Corollary 3.1. Conversely, in order to derive sharp sufficient conditions we require delicate arguments for F satisfying (F1)–(F2). Indeed, the arguments are separated into three cases: (i) $1<p<p_\theta$ (see Theorem 4.1), $p>p_\theta$ (see Theorem 4.3), and (iii) $p=p_\theta$ (see Theorem 4.2). The arguments in case (i) are somewhat standard but the other cases involve certain intricacies, in particular, for the critical case $p=p_\theta$.

The rest of this paper is organized as follows. In Sect. 2, we recall some properties of the fundamental solution $\Gamma _\theta$ and prove some preliminary lemmas. In Sect. 3, we obtain necessary conditions for the existence of local-in-time solutions of problem (P). In Sect. 4, we prove several theorems on sufficient conditions for the existence of local-in-time solutions of problem (P). In Sect. 4.4, we also provide a necessary and sufficient condition on the nonlinearity F for which problem (P) is solvable for the case of initial data a Dirac measure (Corollary 4.4). Finally, in Sect. 5 we complete the proof of our main result, Theorem 1.1, and outline some analogous results for nonlinearities which are asymptotic to further log-refinements of the cases above (see Remark 5.1).

2 Preliminaries

In this section, we prove some important technical lemmas, modifying the arguments in [19] for the more general nonlinearities considered here. We make precise our notion of solution used throughout this paper, which implicitly considers nonnegative functions only. The word ‘solvability’ for problem (P) is always used with respect to this solution concept. In all that follows, we will use C to denote generic positive constants which depend only on N, $\theta$, and F and point out that C may take different values within a calculation. We begin by recalling some properties of the kernel for the fractional Laplacian.

Let $\Gamma _\theta =\Gamma _\theta (x,t)$ be the fundamental solution of

$$\begin{aligned} \partial _t u+(-\Delta )^{\frac{\theta }{2}}u=0\quad \hbox {in}\quad {{ \mathbb {R}}}^N\times (0,\infty ). \end{aligned}$$

The function $\Gamma _\theta$ satisfies

$$\begin{aligned} \begin{aligned}&\Gamma _\theta (x,t)=(4\pi t)^{-\frac{N}{2}}\exp \left( -\frac{|x|^2}{4t}\right) \quad \hbox {if}\quad \theta =2,\\&C^{-1}t^{-\frac{N}{\theta }}\left( 1+t^{-\frac{1}{\theta }}|x|\right) ^{-N-\theta }\le \Gamma _\theta (x,t) \le Ct^{-\frac{N}{\theta }}\left( 1+t^{-\frac{1}{\theta }}|x|\right) ^{-N-\theta }\quad \hbox {if}\quad 0<\theta <2, \end{aligned} \end{aligned}$$

(2.1)

for all $x\in {{ \mathbb {R}}}^N$ and $t>0$ and has the following properties:

$$\begin{aligned}&\bullet&\Gamma _\theta \hbox { is positive and smooth in } {{ \mathbb {R}}}^N\times (0,\infty ) , \nonumber \\&\bullet&\Gamma _\theta (x,t)=t^{-\frac{N}{\theta }}\Gamma _\theta \left( t^{-\frac{1}{\theta }}x,1\right) , \quad \int _{{{\mathbb {R}}}^N}\Gamma _\theta (x,t)\,{\textrm{d}}x=1, \end{aligned}$$

(2.2)

$$\begin{aligned}&\bullet&\Gamma _\theta (\cdot ,1) \hbox { is radially symmetric and }\Gamma _\theta (x,1)\le \Gamma _\theta (y,1) \hbox { if } |x|\ge |y|, \end{aligned}$$

(2.3)

$$\begin{aligned}&\bullet&\Gamma _\theta (x,t)=\int _{ \mathbb {R}^N}\Gamma _\theta (x-y,t-s)\Gamma _\theta (y,s)\,{\textrm{d}}y, \end{aligned}$$

(2.4)

for all $x,y\in {{ \mathbb {R}}}^N$ and $0<s<t$ (see for example [4, 5, 37]). Furthermore, we have the following smoothing estimate for the semigroup associated with $\Gamma _\theta$ (see [19, Lemma 2.1]).

Lemma 2.1

For any $\mu \in {{\mathcal {M}}}$, set

$$\begin{aligned}{}[S(t)\mu ](x):= \int _{{{ \mathbb {R}}}^N} \Gamma _\theta (x-y,t)\,{\textrm{d}}\mu (y), \quad x\in {{ \mathbb {R}}}^N,\,\,\, t>0. \end{aligned}$$

Then, there exists $C=C(N,\theta )>0$ such that

$$\begin{aligned} \Vert S(t)\mu \Vert _{L^\infty ({{ \mathbb {R}}}^N)} \le C t^{-\frac{N}{\theta }} \sup _{x\in {{ \mathbb {R}}}^N} \mu \big (B\big (x,t^\frac{1}{\theta }\big )\big ), \quad t>0. \end{aligned}$$

Remark 2.1

(i) $S(t)\mu$ is possibly infinite everywhere in ${ \mathbb {R}}^N$; (ii) if $\mu \in {{\mathcal {M}}}$ is such that

$$\begin{aligned} \sup _{x\in { \mathbb {R}}^N}\mu (B(x,r ))<\infty \end{aligned}$$

for some $r>0$, then for any $R\ge r$ there exists $C\ge 1$ such that

$$\begin{aligned} \sup _{x\in { \mathbb {R}}^N}\mu (B(x,R))\le C\sup _{x\in { \mathbb {R}}^N}\mu (B(x,r ))<\infty . \end{aligned}$$

See for example [27, Lemma 2.1] or [12, Lemma 2.4].

We now make precise our solution concepts for problem (P).

Definition 2.1

Let $T>0$ and u be a nonnegative, measurable, finite almost everywhere function in $Q_T$. Let F be a nonnegative and continuous function in $[0,\infty )$.

• (i) We say that u satisfies

  $$\begin{aligned} \partial _t u+(-\Delta )^{\frac{\theta }{2}}u=F(u) \end{aligned}$$

  (2.5)

  in $Q_T$ if, for a.a. $\tau \in (0,T)$, u satisfies

  $$\begin{aligned} \qquad u(x,t)=\int _{{{ \mathbb {R}}}^N}\Gamma _\theta (x-y,t-\tau )u(y,\tau )\,{\textrm{d}}y+\int _\tau ^t\int _{{{ \mathbb {R}}}^N}\Gamma _\theta (x-y,t-s)F(u(y,s))\,{\textrm{d}}y\,{\textrm{d}}s \end{aligned}$$

  for a.a. $(x,t)\in {{ \mathbb {R}}}^N\times (\tau ,T)$.

• (ii) Let $\mu \in {{\mathcal {M}}}$. We say that u is a solution of problem (P) in $Q_T$ if u satisfies

  $$\begin{aligned} u(x,t)=\int _{{{ \mathbb {R}}}^N}\Gamma _\theta (x-y,t)\,{\textrm{d}}\mu (y)+\int _0^t\int _{{{ \mathbb {R}}}^N}\Gamma _\theta (x-y,t-s)F(u(y,s))\,{\textrm{d}}y\,{\textrm{d}}s \nonumber \\ \end{aligned}$$

  (2.6)

  for a.a. $(x,t)\in Q_T$. If u satisfies (2.6) with ‘$=$’ replaced by ‘$\ge$,’ then u is said to be a supersolution of problem (P) in $Q_T$.

Next, we recall a lemma on the existence of solutions of problem (P) in the presence of a supersolution (see [19, Lemma 2.2]).

Lemma 2.2

Let F be an increasing, nonnegative continuous function in $[0,\infty )$. Let $\mu \in {{\mathcal {M}}}$ and $0<T\le \infty$. If there exists a supersolution v of problem (P) in $Q_T$, then there exists a solution u of problem (P) in $Q_T$ such that $0\le u(x,t)\le v(x,t)$ in $Q_T$.

Combining Lemma 2.2 and parabolic regularity theory, we have:

Lemma 2.3

Let $\mu \in {{\mathcal {M}}}$ be such that $\displaystyle {\sup _{z\in {{ \mathbb {R}}}^N}}\mu (B(z,1))<\infty$. Suppose

• (i) $F_1$ is nonnegative and locally Lipschitz continuous in $[0,\infty )$;

• (ii) $F_2$ is an increasing and continuous function in $[0,\infty )$ such that $F_1(\tau )\le F_2(\tau )$ for all $\tau \in [0,\infty )$.

If there exists a supersolution v of (P) in $Q_T$ with F replaced by $F_2$ such that for all $\tau \in (0,T)$

$$\begin{aligned} \sup _{\tau<t<T}\Vert v(t)\Vert _{L^\infty ({{ \mathbb {R}}}^N)}<\infty , \end{aligned}$$

(2.7)

then there exists a solution u of (P) in $Q_T$ with F replaced by $F_1$, with u satisfying $0\le u(x,t)\le v(x,t)$ in $Q_T$.

Proof

For any m, $n\in \mathbb {N}$ set

$$\begin{aligned} F_{1,m}(\tau ):= & {} \min \{F_1(\tau ),m\}\quad \hbox { for }\tau \ge 0,\\ \mu _n(x):= & {} \int _{{{ \mathbb {R}}}^N}\Gamma _\theta (x-y,2n^{-1})\,{\textrm{d}}\mu (y)\quad \hbox { for }x\in {{ \mathbb {R}}}^N. \end{aligned}$$

It follows from Lemma 2.1 that $S(n^{-1})\mu \in L^\infty ({{ \mathbb {R}}}^N)$. Also, since $\mu _n=S(n^{-1})S(n^{-1})\mu$, we have that $\mu _n\in BC({{ \mathbb {R}}}^N)$. For each m, $n\in \mathbb {N}$ define the sequence $\{u_{m,n,k}\}_{k=0}^\infty$ by

$$\begin{aligned} \begin{aligned}&u_{m,n,0}(x,t):=\int _{{{ \mathbb {R}}}^N}\Gamma _\theta (x-y,t)\mu _n(y)\,{\textrm{d}}y,\\&u_{m,n,k+1}(x,t):=\int _{{{ \mathbb {R}}}^N}\Gamma _\theta (x-y,t)\mu _n(y)\,{\textrm{d}}y\\&+\int _0^t\int _{{{ \mathbb {R}}}^N}\Gamma _\theta (x-y,t-s)F_{1,m}(u_{m,n,k}(y,s))\,{\textrm{d}}y\,{\textrm{d}}s. \end{aligned} \end{aligned}$$

By (2.4) and Definition 2.1 (ii), we have

$$\begin{aligned} u_{m,n,0}(x,t)&=\int _{{{\mathbb {R}}}^N}\Gamma _\theta (x-y,t)\left( \int _{{{\mathbb {R}}}^N}\Gamma _\theta (y-z,2n^{-1})\,{\textrm{d}}\mu (z)\right) \,{\textrm{d}}y\\&=\int _{{{\mathbb {R}}}^N}\Gamma _\theta (x-z,t+2n^{-1})\,{\textrm{d}}\mu (z)\\&\le v(x,t+2n^{-1}) \end{aligned}$$

for $x\in {{ \mathbb {R}}}^N$ and $t\in [0,T-2n^{-1})$. Since $F_1(\tau )\le F_2(\tau )$ for $\tau \in [0,\infty )$, by induction, we obtain

$$\begin{aligned} 0\le u_{m,n,k}(x,t)\le v(x,t+2n^{-1}) \end{aligned}$$

(2.8)

for all $x\in {{ \mathbb {R}}}^N$, $t\in [0,T-2n^{-1})$, and $k\ge 0$. Here, we used the assumption that $F_2$ is increasing. Since $F_{1,m}$ is globally Lipschitz in $[0,\infty )$, we may apply the standard theory of evolution equations to see that the pointwise limit

$$\begin{aligned} u_{m,n}(x,t):=\lim _{k\rightarrow \infty }u_{m,n,k}(x,t) \end{aligned}$$

exists in ${{ \mathbb {R}}}^N\times [0,\infty )$ and satisfies

$$\begin{aligned} u_{m,n}(x,t)= & {} \int _{{{ \mathbb {R}}}^N}\Gamma _\theta (x-y,t)\mu _n(y)\,{\textrm{d}}y\nonumber \\ {}{} & {} +\int _0^t\int _{{{ \mathbb {R}}}^N}\Gamma _\theta (x-y,t-s)F_{1,m}(u_{m,n}(y,s))\,{\textrm{d}}y\,{\textrm{d}}s \end{aligned}$$

(2.9)

for all $x\in {{ \mathbb {R}}}^N$ and $t>0$. Furthermore, by (2.8) we see that

$$\begin{aligned} 0\le u_{m,n}(x,t)\le v(x,t+2n^{-1}) \end{aligned}$$

(2.10)

for all $x\in {{ \mathbb {R}}}^N$ and $t\in [0,T-2n^{-1})$. Then, by (2.7), for any $\tau \in (0,T-2n^{-1})$ we have

$$\begin{aligned} \sup _{\tau<t<T-2n^{-1}}\Vert u_{m,n}(t)\Vert _{L^\infty ({{ \mathbb {R}}}^N)}\le \sup _{\tau<t<T}\Vert v(t)\Vert _{L^\infty ({{ \mathbb {R}}}^N)}<\infty . \end{aligned}$$

Applying the standard parabolic regularity theory to integral equation (2.9), we find $\alpha \in (0,1)$ such that

$$\begin{aligned} \sup _n \Vert u_{m,n}\Vert _{C^{\alpha ;\alpha /2}(K)}<\infty \end{aligned}$$

(2.11)

for any compact set $K\subset Q_T$. By the Ascoli–Arzelá theorem and the diagonal argument we obtain a subsequence $\{u_{m,n'}\}$ of $\{u_{m,n}\}$ and a function $u_m\in C(Q_T)$ such that

$$\begin{aligned} \lim _{n'\rightarrow \infty }u_{m,n'}(x,t)=u_m(x,t)\quad \hbox {in}\quad Q_T. \end{aligned}$$

(2.12)

Since $F_{1,m}$ is bounded and continuous in $(0,\infty )$, by (2.9), we have

$$\begin{aligned} u_m(x,t)=\int _{{{ \mathbb {R}}}^N}\Gamma _\theta (x-y,t)\mu (y)\,{\textrm{d}}y+\int _0^t\int _{{{ \mathbb {R}}}^N}\Gamma _\theta (x-y,t-s)F_{1,m}(u_m(y,s))\,{\textrm{d}}y\,{\textrm{d}}s\nonumber \\ \end{aligned}$$

(2.13)

in $Q_T$. Furthermore, by (2.10) and (2.12), we see that

$$\begin{aligned} 0\le u_m(x,t)\le v(x,t) \end{aligned}$$

(2.14)

for all $x\in {{ \mathbb {R}}}^N$ and $t\in [0,T)$.

Similarly to (2.11), using (2.14), instead of (2.10), we have

$$\begin{aligned} \sup _m \Vert u_m\Vert _{C^{\alpha ;\alpha /2}(K)}<\infty \end{aligned}$$

for any compact set $K\subset Q_T$. By the Ascoli–Arzelá theorem and the diagonal argument we obtain a subsequence $\{u_{m'}\}$ of $\{u_m\}$ and a function $u\in C(Q_T)$ such that

$$\begin{aligned} \lim _{m'\rightarrow \infty }u_{m'}(x,t)=u(x,t)\quad \hbox {in}\quad Q_T. \end{aligned}$$

(2.15)

Since $F_{1,m}(\tau )\le F_1(\tau )\le F_2(\tau )$ for $\tau \in (0,\infty )$, by (2.14) we see that

$$\begin{aligned}&\sup _{m'}\int _0^t\int _{{{ \mathbb {R}}}^N} \Gamma _\theta (x-y,t-s)F_{1,m'}(u_{m'}(y,s))\,{\textrm{d}}y\,{\textrm{d}}s\\&\le \int _0^t\int _{{{ \mathbb {R}}}^N} \Gamma _\theta (x-y,t-s)F_2(v(y,s))\,{\textrm{d}}y\,{\textrm{d}}s \le v(x,t)<\infty \end{aligned}$$

for a.a. $(x,t)\in Q_T$. Then, by (2.13) and (2.15) we apply Lebesgue’s dominated convergence theorem to see that u is a solution of problem (P) in $Q_T$ with F replaced by $F_1$ and $0\le u(x,t)\le v(x,t)$ in $Q_T$. Thus, Lemma 2.3 follows. $\square$

Next, we provide two lemmas on the relationship between the initial measure and the initial trace for problem (P).

Lemma 2.4

Let F be a nonnegative continuous function in $[0,\infty )$.

• (i) Let u satisfy (2.5) in $Q_T$ for some $T>0$. Then,

  $$\begin{aligned} \underset{0<t<T-\varepsilon }{{\textrm{ess}\; \textrm{sup}}}\,\int _{B(0,R)}u(y,t)\,{\textrm{d}}y<\infty \end{aligned}$$

  for all $R>0$ and $0<\varepsilon <T$. Furthermore, there exists a unique $\nu \in {{\mathcal {M}}}$ as an initial trace of the solution u; that is,

  $$\begin{aligned} \underset{t\rightarrow +0}{{\textrm{ess}\; \textrm{lim}}} \int _{{{ \mathbb {R}}}^N}u(y,t)\eta (y)\,{\textrm{d}}y=\int _{{{ \mathbb {R}}}^N}\eta (y)\,{\textrm{d}}\nu (y) \end{aligned}$$

  for all $\eta \in C_0({{ \mathbb {R}}}^N)$.

• (ii) Let u be a solution of problem (P) in $Q_T$ for some $T>0$. Then, assertion (i) holds with $\nu$ replaced by $\mu$.

The proof of Lemma 2.4 is the same as in [19, Lemma 2.3]. Furthermore, by assertion (i) we can apply the same argument as in the proof of [19, Theorem 1.2] to obtain the following lemma.

Lemma 2.5

Let F be a nonnegative continuous function in $[0,\infty )$ and $T>0$. Let u satisfy (2.5) in $Q_T$. Let $\mu \in {{\mathcal {M}}}$ be the unique initial trace of u guaranteed by Lemma 2.4. If $\sup _{z\in {{ \mathbb {R}}}^N}\mu (B(z,1))<\infty$, then u is a solution of problem (P) in $Q_T$.

In the rest of this section, we prepare preliminary lemmas.

Lemma 2.6

Let $a>0$ and b, $c\in {{\mathbb {R}}}$. Set

$$\begin{aligned} \varphi (\tau ):=\tau ^a(\log \tau )^b(\log \log \tau )^c,\quad \tau \in (e,\infty ). \end{aligned}$$

Then, there exists $L\in (e,\infty )$ such that $\varphi '>0$ in $(L,\infty )$ and the inverse function $\varphi ^{-1}:(\varphi (L),\infty )\rightarrow (L,\infty )$ exists. Furthermore,

$$\begin{aligned} \varphi ^{-1}(\tau )\asymp \tau ^{\frac{1}{a}}(\log \tau )^{-\frac{b}{a}}(\log \log \tau )^{-\frac{c}{a}} \end{aligned}$$

(2.16)

as $\tau \rightarrow \infty$.

Proof

Since $a>0$, we can find $L\in (e,\infty )$ such that

$$\begin{aligned} \varphi '(\tau )= \tau ^{a-1}(\log \tau )^b(\log \log \tau )^c \left[ a+b(\log \tau )^{-1}+c(\log \tau )^{-1}(\log \log \tau )^{-1}\right] >0 \end{aligned}$$

for all $\tau \in (L,\infty )$. Since $\varphi (\tau )\rightarrow \infty$ as $\tau \rightarrow \infty$, it follows that $\varphi ^{-1}:(\varphi (L),\infty )\rightarrow (L,\infty )$ exists and satisfies $\varphi ^{-1}(\tau )\rightarrow \infty$ as $\tau \rightarrow \infty$. Now,

$$\begin{aligned} \log \tau= & {} \log \varphi (\varphi ^{-1}(\tau ))=a\log \varphi ^{-1}(\tau )+b\log \log \varphi ^{-1}(\tau )+c\log \log \log \varphi ^{-1}(\tau )\nonumber \\= & {} a\log \varphi ^{-1}(\tau )(1+o(1)) \end{aligned}$$

(2.17)

as $\tau \rightarrow \infty$, so that

$$\begin{aligned} \log \varphi ^{-1}(\tau )=\frac{1}{a}(\log \tau )(1+o(1)) \end{aligned}$$

as $\tau \rightarrow \infty$. Then, by (2.17) we have

$$\begin{aligned} \begin{aligned}&a\log \varphi ^{-1}(\tau ) =\log \tau -b\log \log \varphi ^{-1}(\tau )-c\log \log \log \varphi ^{-1}(\tau )\\&\quad =\log \tau -b\log \left( \frac{1}{a}(\log \tau )(1+o(1))\right) -c\log \log \left( \frac{1}{a}(\log \tau )(1+o(1))\right) \end{aligned} \end{aligned}$$

as $\tau \rightarrow \infty$. Hence,

$$\begin{aligned} \log \varphi ^{-1}(\tau ) =\log \left[ \left( \tau (\log \tau )^{-b}(\log \log \tau )^{-c}\right) ^{\frac{1}{a}}\right] -\frac{b}{a}\log \left( \frac{1}{a}(1+o(1))\right) +o(1) \end{aligned}$$

as $\tau \rightarrow \infty$, from which (2.16) follows and completes the proof of Lemma 2.6. $\square$

Lemma 2.7

Let $a>0$, $b\ge 0$, and $c\in { \mathbb {R}}$.

• (i) There exists $C_1>0$ such that

  $$\begin{aligned} \int _A^B\tau ^{a-b-1}(\log \tau )^c\,{\textrm{d}}\tau \ge C_1 A^aB^{-b}(\log A)^c\log \frac{B}{A} \end{aligned}$$

  for all A, $B\in [2,\infty )$ with $A\le B$.

• (ii) There exists $C_2\in [1,\infty )$ such that

  $$\begin{aligned}{} & {} C_2^{-1}\tau ^a[\log (e+\tau )]^c\le \int _{\tau /2}^\tau s^{a-1}[\log (e+s)]^c\,{\textrm{d}}s \\{} & {} \le \int _0^\tau s^{a-1}[\log (e+s)]^c\,{\textrm{d}}s \le C_2\tau ^a[\log (e+\tau )]^c \end{aligned}$$

for all $\tau \in [0,\infty )$.

Proof

We first prove assertion (i). Thanks to $a>0$, by Lemma 2.6 we find $R_1\in [2,\infty )$ such that the function $(R_1,\infty )\ni \tau \mapsto \tau ^a(\log \tau )^c$ is increasing. Then, we have

$$\begin{aligned} \left\{ \begin{array}{ll} \tau ^a(\log \tau )^c\ge A^a(\log A)^c &{} \quad \hbox {if}\quad R_1\le A\le \tau ,\\ \tau ^a(\log \tau )^c \ge R_1^a(\log R_1)^c\ge CA^a(\log A)^c &{} \quad \hbox {if}\quad A\le R_1\le \tau ,\\ \tau ^a(\log \tau )^c \ge CA^a(\log A)^c &{} \quad \hbox {if}\quad A\le \tau <R_1,\qquad \end{array} \right. \end{aligned}$$

for all A, $B\in [2,\infty )$ with $A\le B$. We notice that

$$\begin{aligned} \inf _{A\in [2,R_1]}\frac{R_1^a(\log R_1)^c}{A^a(\log A)^c}>0,\quad \inf _{A\in [2,R_1],\,\tau \in [A,R_1]}\frac{\tau ^a(\log \tau )^c}{A^a(\log A)^c}>0. \end{aligned}$$

We observe that

$$\begin{aligned} \begin{aligned} \int _A^B \tau ^{a-b-1}(\log \tau )^c\,{\textrm{d}}\tau&\ge B^{-b} \int _A^B\tau ^{a-1}(\log \tau )^c\,{\textrm{d}}\tau \\&\ge CB^{-b}A^a(\log A)^c\int _A^B\tau ^{-1}\,{\textrm{d}}\tau =CA^aB^{-b}(\log A)^c\log \frac{B}{A} \end{aligned} \end{aligned}$$

for all A, $B\in [2,\infty )$ with $A\le B$. Then, assertion (i) follows.

Next, we prove assertion (ii). Let $\varepsilon \in (0,a)$. By Lemma 2.6 we find $R_2>0$ such that the function $(R_2,\infty )\ni \tau \mapsto (e+\tau )^\varepsilon (\log (e+\tau ))^c$ is increasing. Then, we have

$$\begin{aligned} \int _0^\tau s^{a-1}[\log (e+s)]^c\,{\textrm{d}}s&\le C+\int _{R_2}^\tau s^{a-1}(e+s)^{-\varepsilon }(e+s)^\varepsilon [\log (e+s)]^c\,{\textrm{d}}s\\&\le C+(e+\tau )^\varepsilon [\log (e+\tau )]^c \int _{R_2}^\tau s^{a-1-\varepsilon }\,{\textrm{d}}s\\&\le C+C\tau ^{a-\varepsilon }(e+\tau )^\varepsilon [\log (e+\tau )]^c \le C\tau ^a[\log (e+\tau )]^c \end{aligned}$$

for all $\tau \in [R_2,\infty )$. On the other hand,

$$\begin{aligned} \int _0^\tau s^{a-1}[\log (e+s)]^c\,{\textrm{d}}s \le C\int _0^\tau s^{a-1}\,{\textrm{d}}s\le C\tau ^a\le C\tau ^a[\log (e+\tau )]^c \end{aligned}$$

for all $\tau \in (0,R_2)$. These imply that

$$\begin{aligned} \int _0^\tau s^{a-1}[\log (e+s)]^c\,{\textrm{d}}s\le C\tau ^\varepsilon [\log (e+\tau )]^c\int _0^\tau s^{a-\varepsilon -1}\,{\textrm{d}}s \le C\tau ^a[\log (e+\tau )]^c \nonumber \\ \end{aligned}$$

(2.18)

for all $\tau \in [0,\infty )$. On the other hand, since

$$\begin{aligned} \inf _{\tau \in (0,\infty )}\frac{\log (e+\tau /2)}{\log (e+\tau )}>0, \end{aligned}$$

we have

$$\begin{aligned}{} & {} C^{-1}\log (e+\tau )\le \log (e+\tau /2)\le \inf _{\xi \in (\tau /2,\tau )}\log (e+\xi ) \\{} & {} \quad \le \sup _{\xi \in (\tau /2,\tau )}\log (e+\xi )\le \log (e+\tau ) \end{aligned}$$

for $\tau >0$. This yields

$$\begin{aligned} \int _{\tau /2}^\tau s^{a-1}[\log (e+s)]^c\,{\textrm{d}}s\ge C[\log (e+\tau )]^c\int _{\tau /2}^\tau s^{a-1}\,{\textrm{d}}s \ge C\tau ^a[\log (e+\tau )]^c \nonumber \\ \end{aligned}$$

(2.19)

for all $\tau \in [0,\infty )$. By (2.18) and (2.19), we have assertion (ii). The proof is complete. $\square$

Lemma 2.8

Let $p>1$, $d\in [1,p)$, $q\in {{\mathbb {R}}}$, and $R\ge 0$. Define a function f in $[0,\infty )$ by

$$\begin{aligned} f(\tau ):= \left\{ \begin{array}{cl} 0 &{} \quad \textrm{for}\quad \tau \in [0,R],\\ \displaystyle {\tau ^d\int _R^\tau s^{-d}\left( \int _R^s \xi ^{p-2}[\log (e+\xi )]^q\,{\textrm{d}}\xi \right) \,{\textrm{d}}s} &{} \quad \textrm{for}\quad \tau \in (R,\infty ). \end{array} \right. \end{aligned}$$

Then,

• (i) the function $(0,\infty )\ni \tau \mapsto \tau ^{-d}f(\tau )$ is increasing;

• (ii) f is convex in $[0,\infty )$;

• (iii) $f(\tau )\asymp \tau ^p(\log \tau )^q$ as $\tau \rightarrow \infty$.

Proof

By the definition of f, we easily obtain property (i). Since

$$\begin{aligned} f'(\tau )=d\tau ^{d-1} \int _R^\tau s^{-d}\left( \int _R^s \xi ^{p-2}[\log (e+\xi )]^q\,{\textrm{d}}\xi \right) \,{\textrm{d}}s +\int _{R}^\tau \xi ^{p-2}[\log (e+\xi )]^q\,{\textrm{d}}\xi \end{aligned}$$

for $\tau \in (R,\infty )$, we observe that $f'$ is increasing in $[0,\infty )$, so that property (ii) holds.

We prove property (iii). Since $d\in (1,p)$, by Lemma 2.7 (ii) we have

$$\begin{aligned} \begin{aligned} f(\tau )&\le \displaystyle {\tau ^d\int _0^\tau s^{-d}\left( \int _0^s \xi ^{p-2}[\log (e+\xi )]^q\,{\textrm{d}}\xi \right) \,{\textrm{d}}s}\\&\le C\tau ^d\int _0^\tau s^{p-1-d}[\log (e+s)]^q\,{\textrm{d}}s \le C\tau ^p[\log (e+\tau )]^q \end{aligned} \end{aligned}$$

(2.20)

for all $\tau > R$ and

$$\begin{aligned} \begin{aligned} f(\tau )&\ge \tau ^d\int _{\tau /2}^\tau s^{-d}\left( \int _{s/2}^s \xi ^{p-2}[\log (e+\xi )]^q\,{\textrm{d}}\xi \right) \,{\textrm{d}}s\\&\ge C\tau ^d\int _{\tau /2}^\tau s^{p-1-d}[\log (e+s)]^q\,{\textrm{d}}s \ge C\tau ^p[\log (e+\tau )]^q \end{aligned} \end{aligned}$$

(2.21)

for all $\tau > 4R$. By (2.20) and (2.21), we obtain assertion (iii). Thus, Lemma 2.8 follows. $\square$

3 Necessary conditions for solvability

In this section, we establish necessary conditions for the solvability of problem (P). We begin in Theorem 3.1 by imposing only weak constraints on the nonlinearity F, before specializing to the case where F satisfies (F1) and (F2) in Corollary 3.1.

Theorem 3.1

Let F be a continuous function in $[0,\infty )$. Assume that there exists a convex function f in $[0,\infty )$ with the following properties:

• (f1) $F(\tau )\ge f(\tau )\ge 0$ in $[0,\infty )$;

• (f2) the function $(0,\infty )\ni \tau \mapsto \tau ^{-d}f(\tau )$ is increasing for some $d>1$.

Let u satisfy (2.5) in $Q_T$ for some $T>0$ and let $\mu$ be the initial trace of u. Then, there exists $\gamma =\gamma (N,\theta ,f)\ge 1$ such that

$$\begin{aligned} \int _{\gamma ^{-1}T^{-\frac{N}{\theta }}m_\sigma (z)}^{\gamma ^{-1}\sigma ^{-N}m_\sigma (z)} s^{-p_\theta -1}f(s)\,{\textrm{d}}s\le \gamma ^{p_\theta +1} m_\sigma (z)^{-\frac{\theta }{N}} \end{aligned}$$

(3.1)

for all $z\in {{ \mathbb {R}}}^N$ and $\sigma \in (0,T^{\frac{1}{\theta }})$, where $m_\sigma (z):=\mu (B(z,\sigma ))$.

Proof

It follows from Definition 2.1 (i) and property (f1) that, for a.a. $\tau \in (0,T)$,

$$\begin{aligned} \infty >u(x,t) \ge \int _{{{ \mathbb {R}}}^N}\Gamma _\theta (x-y,t-\tau )u(y,\tau )\,{\textrm{d}}y +\int _\tau ^t\int _{{{ \mathbb {R}}}^N}\Gamma _\theta (x-y,t-s)f(u(y,s))\,{\textrm{d}}y\,{\textrm{d}}s \end{aligned}$$

for a.a. $x\in {{ \mathbb {R}}}^N$ and a.a. $t\in (\tau ,T)$. This implies that

$$\begin{aligned} \infty >u(x,2t)\ge \int _{{{ \mathbb {R}}}^N}\Gamma _\theta (x-y,t)u(y,t)\,{\textrm{d}}y \end{aligned}$$

(3.2)

for a.a. $x\in {{ \mathbb {R}}}^N$ and a.a. $t\in (0,T/2)$.

Let $0<\rho <(T/2)^{\frac{1}{\theta }}$. It follows from Definition 2.1 (i), property (f1), and (2.4) that

$$\begin{aligned} \begin{aligned}&\int _{{{ \mathbb {R}}}^N}\Gamma _\theta (z-x,t)u(x,t)\,{\textrm{d}}x\\&\quad \ge \int _{{{ \mathbb {R}}}^N}\int _{{{ \mathbb {R}}}^N}\Gamma _\theta (z-x,t)\Gamma _\theta (x-y,t)\,{\textrm{d}}\mu (y)\,{\textrm{d}}x\\&\qquad +\int _0^t\int _{{{ \mathbb {R}}}^N}\int _{{{ \mathbb {R}}}^N}\Gamma _\theta (z-x,t)\Gamma _\theta (x-y,t-s)f(u(y,s))\,{\textrm{d}}y\,{\textrm{d}}s\,{\textrm{d}}x\\&\quad =\int _{{{ \mathbb {R}}}^N}\Gamma _\theta (z-y,2t)\,{\textrm{d}}\mu (y) +\int _0^t\int _{{{ \mathbb {R}}}^N}\Gamma _\theta (z-y,2t-s)f(u(y,s))\,{\textrm{d}}y\,{\textrm{d}}s \end{aligned} \end{aligned}$$

(3.3)

for all $z\in {{ \mathbb {R}}}^N$ and a.a. $t\in (0,T)$. On the other hand, by (2.1) we have

$$\begin{aligned} \int _{{{ \mathbb {R}}}^N}\Gamma _\theta (z-y,2t)\,{\textrm{d}}\mu (y)\ge & {} \int _{B(z,\sigma )}\Gamma _\theta (z-y,2t)\,{\textrm{d}}\mu (y) \nonumber \\\ge & {} \min _{x\in B(0,\sigma )}\Gamma _\theta (x,2t) \mu (B(z,\sigma )) \ge Ct^{-\frac{N}{\theta }}\mu (B(z,\sigma ))\qquad \quad \end{aligned}$$

(3.4)

for all $z\in {{ \mathbb {R}}}^N$ and $t\ge \rho ^\theta$, where $\sigma :=2^{\frac{1}{\theta }}\rho \in (0,T^{\frac{1}{\theta }})$. Furthermore, by (2.2) and (2.3) we see that

$$\begin{aligned} \Gamma _\theta (z-y,2t-s)= & {} (2t-s)^{-\frac{N}{\theta }}\Gamma _\theta \left( (2t-s)^{-\frac{1}{\theta }}(z-y),1\right) \nonumber \\\ge & {} \left( \frac{s}{2t}\right) ^{\frac{N}{\theta }}s^{-\frac{N}{\theta }}\Gamma _\theta \left( s^{-\frac{1}{\theta }}(z-y),1\right) =\left( \frac{s}{2t}\right) ^{\frac{N}{\theta }}\Gamma _\theta (z-y,s)\qquad \quad \end{aligned}$$

(3.5)

for all y, $z\in {{ \mathbb {R}}}^N$ and $0<s<t$. Combining (3.2), (3.3), (3.4), and (3.5), we find $C_*\ge 1$ such that

$$\begin{aligned} \begin{aligned}&\infty >w(t):= \int _{{{ \mathbb {R}}}^N}\Gamma _\theta (z-x,t)u(x,t)\,{\textrm{d}}x\\&\quad \ge \, C_*^{-1}t^{-\frac{N}{\theta }}\mu (B(z,\sigma )) +C_*^{-1}t^{-\frac{N}{\theta }}\int _{\rho ^\theta }^t\int _{{{ \mathbb {R}}}^N}s^{\frac{N}{\theta }}\Gamma _\theta (z-y,s)f(u(y,s))\,{\textrm{d}}y\,{\textrm{d}}s \end{aligned} \end{aligned}$$

for a.a. $z\in {{ \mathbb {R}}}^N$ and a.a. $t\in (\rho ^\theta ,T/2)$. Thanks to the convexity of f, by (2.2) we may apply Jensen’s inequality to obtain

$$\begin{aligned} \infty >w(t)\ge & {} C_*^{-1}t^{-\frac{N}{\theta }}\mu (B(z,\sigma )) +C_*^{-1}t^{-\frac{N}{\theta }}\int _{\rho ^\theta }^t s^{\frac{N}{\theta }} f\left( \int _{{{ \mathbb {R}}}^N} \Gamma _\theta (z-y,s)u(y,s)\,{\textrm{d}}y\right) \,{\textrm{d}}s\nonumber \\= & {} C_*^{-1}t^{-\frac{N}{\theta }}\mu (B(z,\sigma )) +C_*^{-1}t^{-\frac{N}{\theta }}\int _{\rho ^\theta }^t s^{\frac{N}{\theta }}f(w(s))\,{\textrm{d}}s \end{aligned}$$

(3.6)

for a.a. $z\in {{ \mathbb {R}}}^N$ and a.a. $t\in (\rho ^\theta ,T/2)$.

Since f is convex in $[0,\infty )$, it is Lipschitz continuous in any compact subinterval of $[0,\infty )$. We may then let $\zeta$ denote the unique local solution of the integral equation

$$\begin{aligned} \zeta (t)=\mu (B(z,\sigma )) +\int _{\rho ^\theta }^t s^{\frac{N}{\theta }}f(C_*^{-1}s^{-\frac{N}{\theta }}\zeta (s))\,{\textrm{d}}s,\quad t\ge \rho ^\theta . \end{aligned}$$

(3.7)

Hence, $\zeta$ is the unique local solution of

$$\begin{aligned} \zeta '(t)=t^{\frac{N}{\theta }}f\left( C_*^{-1}t^{-\frac{N}{\theta }}\zeta (t)\right) , \qquad \zeta (\rho ^\theta )=\mu (B(z,\sigma )). \end{aligned}$$

(3.8)

By (3.6), applying the standard theory for ordinary differential equations to (3.7), we see that the solution $\zeta$ exists in $[\rho ^\theta ,T/2)$ and satisfies

$$\begin{aligned} \zeta (t)\le C_* t^{\frac{N}{\theta }}w(t)<\infty ,\quad t\in [\rho ^\theta ,T/2). \end{aligned}$$

It follows from (3.8) and property (f2) that

$$\begin{aligned} \begin{aligned} \zeta '(t)&=t^{\frac{N}{\theta }} \left[ C_*^{-1}t^{-\frac{N}{\theta }}\zeta (t)\right] ^d \left[ C_*^{-1}t^{-\frac{N}{\theta }}\zeta (t)\right] ^{-d} f\left( C_*^{-1}t^{-\frac{N}{\theta }}\zeta (t)\right) \\&\ge t^{\frac{N}{\theta }} \left[ C_*^{-1}t^{-\frac{N}{\theta }}\zeta (t)\right] ^d\left[ C_*^{-1}t^{-\frac{N}{\theta }}\zeta (\rho ^\theta )\right] ^{-d} f\left( C_*^{-1}t^{-\frac{N}{\theta }}\zeta (\rho ^\theta )\right) \\&\ge \zeta (\rho ^\theta )^{-d}t^{\frac{N}{\theta }}\zeta (t)^d f\left( C_*^{-1}t^{-\frac{N}{\theta }}\zeta (\rho ^\theta )\right) \end{aligned} \end{aligned}$$

for all $t\in [\rho ^\theta ,T/2)$. Then, we have

$$\begin{aligned} \frac{1}{d-1}\zeta (\rho ^\theta )^{1-d} \ge \int _{\rho ^\theta }^{T/2} \frac{\zeta '(s)}{\zeta (s)^d}\,{\textrm{d}}s \ge \zeta (\rho ^\theta )^{-d} \int _{\rho ^\theta }^{T/2} s^{\frac{N}{\theta }}f\left( C_*^{-1}s^{-\frac{N}{\theta }}\zeta (\rho ^\theta )\right) \,{\textrm{d}}s. \end{aligned}$$

Recalling (3.8) and setting $\eta :=C_*^{-1}\mu (B(z,\sigma )) s^{-\frac{N}{\theta }}$, we take large enough $C_*$ if necessary so that

$$\begin{aligned} \begin{aligned} \mu (B(z,\sigma ))&\ge (d-1)\int _{\rho ^\theta }^{T/2} s^{\frac{N}{\theta }}f\left( C_*^{-1}s^{-\frac{N}{\theta }}\zeta (\rho ^\theta )\right) \,{\textrm{d}}s\\&=(d-1)\int _{\rho ^\theta }^{T/2} s^{\frac{N}{\theta }}f\left( C_*^{-1}\mu (B(z,\sigma ))s^{-\frac{N}{\theta }}\right) \,{\textrm{d}}s\\&=\frac{(d-1)\theta }{N}C_*^{-p_\theta }\mu (B(z,\sigma ))^{p_\theta }\int _{C_*^{-1}(T/2)^{-\frac{N}{\theta }}\mu (B(z,\sigma ))}^{C_*^{-1}\rho ^{-N}\mu (B(z,\sigma ))} \eta ^{-p_\theta -1}f(\eta )\,{\textrm{d}}\eta \\&\ge \gamma ^{-p_\theta -1}\mu (B(z,\sigma ))^{p_\theta }\int _{\gamma ^{-1}T^{-\frac{N}{\theta }}\mu (B(z,\sigma ))}^{\gamma ^{-1}\sigma ^{-N}\mu (B(z,\sigma ))} \eta ^{-p_\theta -1}f(\eta )\,{\textrm{d}}\eta \end{aligned} \end{aligned}$$

for all $z\in {\mathbb {R}}^N$ and $\sigma \in (0,T^{\frac{1}{\theta }})$, where $\gamma =2^{-\frac{N}{\theta }}C_*$. Here, we used the relation $\sigma =2^{\frac{1}{\theta }}\rho \in (0,T^{\frac{1}{\theta }})$. Thus, inequality (3.1) holds, and the proof of Theorem 3.1 is complete. $\square$

Corollary 3.1

Assume conditions (F1) and (F2). Let u satisfy

$$\begin{aligned} \partial _t u+(-\Delta )^{\frac{\theta }{2}}u=F(u) \end{aligned}$$

in $Q_T$ for some $T>0$. Then, there exists a unique $\nu \in {{\mathcal {M}}}$ as the initial trace of u. Furthermore,

• (i) u is a solution of problem (P) in $Q_T$ with $\mu =\nu$;

• (ii) there exists $C=C(N,\theta ,F)>0$ such that

  $$\begin{aligned} \sup _{z\in {{ \mathbb {R}}}^N}\mu (B(z,\sigma ))\le \left\{ \begin{array}{ll} C\sigma ^{N-\frac{\theta }{p-1}}|\log \sigma |^{-\frac{q}{p-1}} &{}\quad \text {if}\quad p\ne p_\theta ,\\ C|\log \sigma |^{-\frac{N(q+1)}{\theta }} &{}\quad \text {if}\quad p=p_\theta ,\,\,\,q\ne -1,\\ C[\log |\log \sigma |]^{-\frac{N}{\theta }} &{}\quad \text {if}\quad p=p_\theta ,\,\,\,q=-1, \end{array} \right. \end{aligned}$$

  for all small enough $\sigma >0$.

Proof

The existence and uniqueness of the initial trace of u follows from Lemma 2.4. Let $d\in (1,p)$, $R>0$, and $\kappa >0$. Set

$$\begin{aligned} f(\tau ):= \left\{ \begin{array}{ll} 0 &{} \quad \hbox {for}\quad 0\le \tau <R,\\ \displaystyle {\kappa \tau ^d\int _R^\tau s^{-d}\left( \int _R^s \xi ^{p-2}[\log (e+\xi )]^q\,{\textrm{d}}\xi \right) \,{\textrm{d}}s} &{} \quad \hbox {for}\quad \tau \ge R. \end{array} \right. \end{aligned}$$

By Lemma 2.8 (i) and (ii), we see that f is convex in $(0,\infty )$ and (f2) in Theorem 3.1 holds. Furthermore, thanks to Lemma 2.8 (iii), taking small enough $\kappa >0$ and large enough $R>0$, by (F2) we can ensure that $F(\tau )\ge f(\tau )$ in $[0,\infty )$ and consequently (f1) in Theorem 3.1 also holds. In particular, we find $L\in (R,\infty )$ such that

$$\begin{aligned} F(\tau )\ge f(\tau )\ge C\tau ^p(\log \tau )^q,\quad \tau \in (L,\infty ). \end{aligned}$$

(3.9)

By Theorem 3.1, we also find $\gamma \ge 1$ such that

$$\begin{aligned} \gamma ^{p_\theta +1} m_\sigma (z)^{-\frac{\theta }{N}}\ge \int _{\gamma ^{-1}T^{-\frac{N}{\theta }}m_\sigma (z)}^{\gamma ^{-1}\sigma ^{-N}m_\sigma (z)}s^{-p_\theta -1}f(s)\,{\textrm{d}}s \end{aligned}$$

(3.10)

for all $z\in \mathbb {R}^N$ and $\sigma \in (0,T^{\frac{1}{\theta }})$.

We show that

$$\begin{aligned} \sup _{z\in {{ \mathbb {R}}}^N}m_\sigma (z)<\infty \quad \hbox {for all }\sigma \in \big (0,T^{\frac{1}{\theta }}\big ). \end{aligned}$$

(3.11)

For then by Remark 2.1 (ii), we have

$$\begin{aligned} \sup _{z\in {{ \mathbb {R}}}^N}\mu (B(z,1))\le C\sup _{z\in {{ \mathbb {R}}}^N}\mu (B(z,T^{\frac{1}{\theta }}/2))=C\sup _{z\in {{ \mathbb {R}}}^N}m_{T^{\frac{1}{\theta }}/2}(z)<\infty , \end{aligned}$$

(3.12)

and assertion (i) will follow from Lemma 2.5.

Suppose that $\sigma \in (0,T^{\frac{1}{\theta }})$ but (3.11) does not hold. Then, there exists a sequence $\{z_n\}\subset \mathbb {R}^N$ such that $m_\sigma (z_n)\rightarrow \infty$ as $n\rightarrow \infty$. Consequently,

$$\begin{aligned} \gamma ^{-1}T^{-\frac{N}{\theta }}m_\sigma (z_n)\ge \max \{ L,2\} \end{aligned}$$

(3.13)

for all n large enough. By (3.9), (3.10), (3.13), and Lemma 2.7 (i) (with $a=p-1$, $b=\theta /N$, and $c=q$), we obtain

$$\begin{aligned} \begin{aligned}&m_\sigma (z_n)^{-\frac{\theta }{N}} \ge C\gamma ^{-p_\theta -1}\int _{\gamma ^{-1}T^{-\frac{N}{\theta }}m_\sigma (z_n)}^{\gamma ^{-1}\sigma ^{-N}m_\sigma (z_n)} s^{p-p_\theta -1}(\log s)^q\,{\textrm{d}}s\\&\ge C\left( \gamma ^{-1}T^{-\frac{N}{\theta }}m_\sigma (z_n)\right) ^{p-1} \left( \gamma ^{-1}\sigma ^{-N}m_\sigma (z_n)\right) ^{-\frac{\theta }{N}} \left( \log \left( \gamma ^{-1}T^{-\frac{N}{\theta }}m_\sigma (z_n)\right) \right) ^q\log \left( \frac{\sigma ^{-N}}{T^{-\frac{N}{\theta }}}\right) \\&=C\sigma ^\theta T^{-\frac{N(p-1)}{\theta }}\log \left( {\sigma ^{-N}}{T^{\frac{N}{\theta }}}\right) m_\sigma (z_n)^{p-1-\frac{\theta }{N}} \left( \log \left( \gamma ^{-1}T^{-\frac{N}{\theta }}m_\sigma (z_n)\right) \right) ^q. \end{aligned} \end{aligned}$$

Hence,

$$\begin{aligned} 1\ge C\sigma ^\theta T^{-\frac{N(p-1)}{\theta }}\log \left( {\sigma ^{-N}}{T^{\frac{N}{\theta }}}\right) m_\sigma (z_n)^{p-1} \left( \log \left( \gamma ^{-1}T^{-\frac{N}{\theta }}m_\sigma (z_n)\right) \right) ^q.\qquad \end{aligned}$$

(3.14)

Letting $n\rightarrow \infty$ in (3.14) yields a contradiction and thus (3.11) holds.

We now prove assertion (ii). Consider first the case where $p\not =p_\theta$. We show that there exist $C>0$ and $\sigma _*>0$ such that

$$\begin{aligned} \sigma ^{\frac{\theta }{p-1}-N}|\log \sigma |^{\frac{q}{p-1}}m_\sigma (z)\le C \end{aligned}$$

(3.15)

for all $z\in { \mathbb {R}}^N$ and $\sigma \in (0,\sigma _*)$. Suppose, for contradiction, that there exist sequences $\{z_n\}\subset \mathbb {R}^N$ and $\{\sigma _n\}\subset (0,\infty )$ such that

$$\begin{aligned} \sigma _n\rightarrow 0\qquad \text {and}\qquad \sigma _n^{\frac{\theta }{p-1}-N}|\log \sigma _n|^{\frac{q}{p-1}}m_{\sigma _n}(z_n)\rightarrow \infty \qquad \text {as}\qquad n\rightarrow \infty . \end{aligned}$$

(3.16)

Set $M_n:=m_{\sigma _n}(z_n)$. It follows from (3.12) that

$$\begin{aligned} M_n\le \sup _{z\in {{ \mathbb {R}}}^N}\mu (B(z,1))\le C \end{aligned}$$

(3.17)

for all n large enough. By (3.16), we necessarily have

$$\begin{aligned} \sigma _n^{-N}M_n\rightarrow \infty \qquad \text {as}\qquad n\rightarrow \infty , \end{aligned}$$

so that for n large enough,

$$\begin{aligned} \gamma ^{-1}(2\sigma _n)^{-N}M_n\ge \max \{ L,2\}\qquad \text {and}\qquad (2\sigma _n)^{-N}>T^{-\frac{N}{\theta }}. \end{aligned}$$

(3.18)

Similar to the proof of part (i), it follows from (3.9), (3.10), and (3.18) that

$$\begin{aligned} \gamma ^{p_\theta +1} \sigma _n^{-\theta } \ge C\left( \sigma _n^{-N}M_n\right) ^{\frac{\theta }{N}}\int _{\gamma ^{-1}(2\sigma _n)^{-N}M_n}^{\gamma ^{-1}\sigma _n^{-N}M_n}s^{p-p_\theta -1}(\log s)^q\,{\textrm{d}}s. \end{aligned}$$

Applying Lemma 2.7 (i) (with $a=p-1$, $b=\theta /N$, and $c=q$), we obtain (for n large enough)

$$\begin{aligned} C\sigma _n^{-\theta }\ge \tau _n^{p-1}(\log (C\tau _n))^q, \end{aligned}$$

(3.19)

where $\tau _n:=\sigma _n^{-N}M_n$. For n large enough, and rescaling with $s_n=C\tau _n$ in (3.19), we can apply Lemma 2.6 (with $a=p-1>0$, $b=q$, and $c=0$) to obtain (after rescaling back to $\tau _n$)

$$\begin{aligned} \sigma _n^{-N}M_n=\tau _n \le C\left( C\sigma _n^{-\theta }\right) ^{\frac{1}{p-1}}\left( \log \left[ C\sigma _n^{-\theta }\right] \right) ^{-\frac{q}{p-1}}. \end{aligned}$$

Consequently, for such n,

$$\begin{aligned} \sigma _n^{\frac{\theta }{p-1}-N}|\log \sigma _n|^{\frac{q}{p-1}}M_n\le C, \end{aligned}$$

contradicting (3.16). Thus, (3.15) holds, as required.

Now consider the case when $p=p_\theta$ and $q\not =-1$. We show that there exist $C>0$ and $\sigma _*>0$ such that

$$\begin{aligned} |\log \sigma |^{\frac{N(q+1)}{\theta }}m_\sigma (z)\le C \end{aligned}$$

(3.20)

for all $z\in { \mathbb {R}}^N$ and $\sigma \in (0,\sigma _*)$. Suppose, for contradiction, that there exist sequences $\{z_n\}\subset \mathbb {R}^N$ and $\{\sigma _n\}\subset (0,\infty )$ such that

$$\begin{aligned} \sigma _n\rightarrow 0\qquad \text {and}\qquad |\log \sigma _n|^{\frac{N(q+1)}{\theta }}m_{\sigma _n}(z_n) \rightarrow \infty \qquad \text {as}\qquad n\rightarrow \infty . \end{aligned}$$

(3.21)

Set $M_n:=m_{\sigma _n}(z_n)$. Since $\sigma ^{-N/2}\ge |\log \sigma |^{\frac{N(q+1)}{\theta }}$ for all $\sigma >0$ small enough, by (3.21), we necessarily have

$$\begin{aligned} \sigma _n^{-\frac{N}{2}}M_n\rightarrow \infty \qquad \text {as}\qquad n\rightarrow \infty , \end{aligned}$$

so that for n large enough,

$$\begin{aligned} \gamma ^{-1}\sigma _n^{-\frac{N}{2}}M_n\ge \max \{ L,2\}\qquad \text {and}\qquad \sigma _n^{-\frac{N}{2}}>T^{-\frac{N}{\theta }}. \end{aligned}$$

(3.22)

Similar to the proof of part (i), it follows from (3.9), (3.10), and (3.22) that

$$\begin{aligned} \gamma ^{p_\theta +1} \ge CM_n^{\frac{\theta }{N}}\int _{\gamma ^{-1}\sigma _n^{-\frac{N}{2}}M_n}^{\gamma ^{-1}\sigma _n^{-N}M_n}s^{-1}(\log s)^q\,{\textrm{d}}s. \end{aligned}$$

(3.23)

Now set $c_q:=1/2$ if $q\ge 0$ and $c_q:=1$ if $q<0$. Then, by (3.23) we have

$$\begin{aligned} \begin{aligned} 1&\ge CM_n^{\frac{\theta }{N}} \left( \log (\gamma ^{-1}\sigma _n^{-Nc_q} M_n)\right) ^q\int _{\gamma ^{-1}\sigma _n^{-\frac{N}{2}}M_n} ^{\gamma ^{-1}\sigma _n^{-N}M_n}\tau ^{-1}\,{\textrm{d}}\tau \\&= CM_n^{\frac{\theta }{N}}\left( \log \left( \gamma ^{-1}\sigma _n^{-Nc_q}M_n\right) \right) ^q\log \left( \sigma _n^{-\frac{N}{2}}\right) , \end{aligned} \end{aligned}$$

so that

$$\begin{aligned} \left( \sigma _n^{-Nc_q} M_n\right) ^{\frac{\theta }{N}} \left( \log \left( \gamma ^{-1}\sigma _n^{-Nc_q}M_n\right) \right) ^q \le C\sigma _n^{-c_q\theta }|\log \sigma _n|^{-1}. \end{aligned}$$

(3.24)

Setting $\tau _n:=\gamma ^{-1}\sigma _n^{-Nc_q}M_n$, (3.24) can be written as

$$\begin{aligned} \tau _n^{\frac{\theta }{N}}\left( \log \tau _n\right) ^q \le C\sigma _n^{-c_q\theta }|\log \sigma _n|^{-1}. \end{aligned}$$

(3.25)

Applying Lemma 2.6 to (3.25) (with $a=\theta /N$, $b=q$, and $c=0$) then yields

$$\begin{aligned} \begin{aligned} \sigma _n^{-Nc_q}M_n&\le C\left( \sigma _n^{-c_q\theta }|\log \sigma _n|^{-1}\right) ^{\frac{N}{\theta }} \left( \log \left( C\sigma _n^{-c_q\theta }|\log \sigma _n|^{-1}\right) \right) ^{-\frac{Nq}{\theta }}\\&\le C\sigma _n^{-Nc_q}|\log \sigma _n|^{-\frac{N}{\theta }} \left( C\log \left( \sigma _n^{-1}\right) \right) ^{-\frac{Nq}{\theta }}\\&\le C\sigma _n^{-Nc_q}|\log \sigma _n|^{-\frac{N(q+1)}{\theta }} \end{aligned} \end{aligned}$$

for all n large enough. Consequently for such n,

$$\begin{aligned} |\log \sigma _n|^{\frac{N(q+1)}{\theta }}M_n\le C, \end{aligned}$$

contradicting (3.21). Thus, (3.20) holds, as required.

Finally, consider the case when $p=p_\theta$ and $q=-1$. We show that there exist $C>0$ and $\sigma _*>0$ such that

$$\begin{aligned} \left( \log |\log \sigma |\right) ^{\frac{N}{\theta }}m_\sigma (z)\le C \end{aligned}$$

(3.26)

for all $z\in { \mathbb {R}}^N$ and $\sigma \in (0,\sigma _*)$. Suppose, for contradiction, that there exist sequences $\{z_n\}\subset \mathbb {R}^N$ and $\{\sigma _n\}\subset (0,\infty )$ such that

$$\begin{aligned} \sigma _n\rightarrow 0\qquad \text {and}\qquad \left( \log |\log \sigma _n|\right) ^{\frac{N}{\theta }}m_{\sigma _n}(z_n) \rightarrow \infty \qquad \text {as}\qquad n\rightarrow \infty . \end{aligned}$$

(3.27)

Set $M_n:=m_{\sigma _n}(z_n)$. Since $\sigma ^{-N}\ge \left( \log |\log \sigma |\right) ^{\frac{N}{\theta }}$ for all $\sigma >0$ small enough, by (3.27), we necessarily have

$$\begin{aligned} \sigma _n^{-N}M_n\rightarrow \infty \qquad \text {as}\qquad n\rightarrow \infty . \end{aligned}$$

Then, combining (3.17), we find $L'>0$ such that

$$\begin{aligned} \max \{\gamma ^{-1}T^{-\frac{N}{\theta }}M_n, L,2\}\le L'<\gamma ^{-1}\sigma _n^{-N}M_n \end{aligned}$$

(3.28)

for all n large enough. Once again, by (3.9), (3.10), and (3.28) we have

$$\begin{aligned} \begin{aligned} \sigma _n^{-\theta }&\ge C\gamma ^{-p_\theta -1}\left( \sigma _n^{-N}M_n\right) ^{\frac{\theta }{N}} \int _{L'}^{\gamma ^{-1}\sigma _n^{-N}M_n} \tau ^{-1}(\log \tau )^{-1}\,{\textrm{d}}\tau \\&= C \tau _n^{\frac{\theta }{N}}\log \left( \frac{\log \tau _n}{\log L'}\right) \ge C\tau _n^{\frac{\theta }{N}}\log \log \tau _n \end{aligned} \end{aligned}$$

for all n large enough, where $\tau _{n}:=\gamma ^{-1}\sigma _n^{-N}M_n$. By (3.22) and Lemma 2.6 (with $a=\theta /N$, $b=0$, and $c=1$), we have

$$\begin{aligned} \gamma ^{-1}\sigma _n^{-N}M_n=\tau _{n}\le \left( C\sigma _n^{-\theta }\right) ^{\frac{N}{\theta }} \left( \log \log \left[ C\sigma _n^{-\theta }\right] \right) ^{-\frac{N}{\theta }} \le C\sigma _n^{-N}\left( \log |\log \sigma _n|\right) ^{-\frac{N}{\theta }}, \end{aligned}$$

so that

$$\begin{aligned} \left( \log |\log \sigma _n|\right) ^{\frac{N}{\theta }}M_n\le C, \end{aligned}$$

contradicting (3.27). Hence, (3.26) holds, as required. The proof of Corollary 3.1 is complete. $\square$

4 Sufficient conditions for solvability

In this section, we establish sufficient conditions for the existence of a supersolution, and consequently of a local-in-time solution of problem (P), for three general classes of nonlinearity F (see Theorems 4.1, 4.2, and 4.3). As corollaries, we obtain the corresponding results when specializing to nonlinearities satisfying (F1) and (F2) (Corollaries 4.1, 4.2, and 4.3). Indeed, for F satisfying (F1) and (F2) the classification of initial data for which problem (P) is locally solvable separates naturally into the following three cases:

• (A): $\hbox {either}\quad {\mathrm{(i)}}\quad 1<p<p_\theta \hbox { and } q\in {{\mathbb {R}}}\quad \hbox {or}\quad {\mathrm{(ii)}}\quad { p=p_\theta \hbox { and } q<-1};$

• (B): $p=p_\theta$ and $q\ge -1$;

• (C): $p>p_\theta$.

4.1 Sufficiency: case (A)

We begin with nonlinearities F which generalize case (A).

Theorem 4.1

Let F be a nonnegative continuous function in $[0,\infty )$ and assume the following conditions:

• (A1) there exists $R\ge 0$ such that the function $(R,\infty )\ni \tau \mapsto \tau ^{-1}F(\tau )$ is increasing;

• (A2) $\displaystyle {\int _1^\infty \tau ^{-p_\theta -1}F(\tau )\,{\textrm{d}}\tau <\infty }$.

If $\mu \in {{\mathcal {M}}}$ satisfies

$$\begin{aligned} \sup _{x\in {{ \mathbb {R}}}^N}\mu (B(x,1))<\infty , \end{aligned}$$

(4.1)

then problem (P) possesses a solution u in $Q_T$ for some $T>0$, with u satisfying

$$\begin{aligned} 0\le u(x,t)\le 2[S(t)\mu ](x)+R\le Ct^{-\frac{N}{\theta }} \end{aligned}$$

in $Q_T$ for some $C>0$.

Proof

Let $T\in (0,1)$ be chosen later. Set $w(x,t):=R+2[S(t)\mu ](x)$. It follows from Lemma 2.1 and (4.1) that

$$\begin{aligned} R\le w(x,t)\le R+Ct^{-\frac{N}{\theta }}\sup _{z\in {{ \mathbb {R}}}^N}\mu \big (B\big (z,t^{\frac{1}{\theta }}\big )\big ) \le R+Mt^{-\frac{N}{\theta }}\le 2Mt^{-\frac{N}{\theta }} \end{aligned}$$

(4.2)

for $0<t\le T$ and small enough T, where $M:=C{\sup _{x\in {{ \mathbb {R}}}^N}}\mu (B(x,1))+1<\infty$. Then, by (A1) and (4.2) we have

$$\begin{aligned} \begin{aligned} 0\le \frac{F(w(x,t))}{w(x,t)} \le (2M)^{-1}t^{\frac{N}{\theta }}F\big (2Mt^{-\frac{N}{\theta }}\big ), \quad (x,t)\in Q_T. \end{aligned} \end{aligned}$$

(4.3)

Noting that

$$\begin{aligned} S(t-s)w(s)=S(t-s)[R+2S(s)\mu ]=R+2S(t)\mu =w(t), \end{aligned}$$

then by (A2) and (4.3) we obtain

$$\begin{aligned} \begin{aligned}&[S(t)\mu ](x)+\int _0^t S(t-s)F(w(s))\,{\textrm{d}}s\\&\le \frac{1}{2}w(x,t)+\int _0^t \left\| \frac{F(w(s))}{w(s)}\right\| _{L^\infty ({{ \mathbb {R}}}^N)} S(t-s)w(s)\,{\textrm{d}}s\\&\le \frac{1}{2}w(x,t)+(2M)^{-1}w(x,t) \int _0^t s^{\frac{N}{\theta }}F\left( 2Ms^{-\frac{N}{\theta }}\right) \,{\textrm{d}}s\\&\le w(x,t) \left[ \frac{1}{2}+CM^{\frac{\theta }{N}} \int _{2MT^{-\frac{N}{\theta }}}^\infty \tau ^{-p_\theta -1}F(\tau )\,{\textrm{d}}\tau \right] \le w(x,t), \quad (x,t)\in Q_T, \end{aligned} \end{aligned}$$

for small enough T. This means that w is a supersolution in $Q_T$ and the desired result follows from Lemma 2.2 and (4.2). $\square$

Corollary 4.1

Assume conditions (F1) and (F2) with

$$\begin{aligned} \hbox {either}\quad {\mathrm{(i)}}\quad {1<p<p_\theta \hbox { and }q\in {{ \mathbb {R}}}}\qquad \hbox {or}\qquad {\mathrm{(ii)}}\quad {p=p_\theta \hbox { and }q<-1}. \end{aligned}$$

If $\mu \in {{\mathcal {M}}}$ satisfies

$$\begin{aligned} \sup _{z\in {{\mathbb {R}}}^N}\mu (B(z,1))<\infty , \end{aligned}$$

then problem (P) possesses a solution u in $Q_T$ for some $T>0$, with u satisfying

$$\begin{aligned} 0\le u(x,t)\le 2[S(t)\mu ](x)+R\le Ct^{-\frac{N}{\theta }} \end{aligned}$$

in $Q_T$ for some $R>0$ and $C>0$.

Proof

Set

$$\begin{aligned} g(\tau ):=\tau \int _0^\tau s^{-1}\left( \int _0^s \xi ^{p-2}[\log (e+\xi )]^q\,{\textrm{d}}\xi \right) \,{\textrm{d}}s, \qquad \tau \ge 0. \end{aligned}$$

It follows from Lemma 2.8 (iii) (with $d=1$ and $R=0$) that $g(\tau )\asymp \tau ^p[\log \tau ]^q$ as $\tau \rightarrow \infty$. Hence, since either $1<p<p_\theta$, or $p=p_\theta$ and $q<-1$, we have

$$\begin{aligned} \int _1^\infty \tau ^{-p_\theta -1}g(\tau )\,{\textrm{d}}\tau \le C \int _1^\infty \tau ^{p-p_\theta -1}[\log \tau ]^q\,{\textrm{d}}\tau <\infty . \end{aligned}$$

(4.4)

Let $\kappa >0$ and $L>0$. Set

$$\begin{aligned} f(\tau ):=\kappa g(\tau )+L , \qquad \tau \ge 0. \end{aligned}$$

(4.5)

Clearly, $f(\tau )\asymp g(\tau )\asymp \tau ^p[\log \tau ]^q$ as $\tau \rightarrow \infty$ and so by (F1)–(F2) we may choose $\kappa$ and L large enough such that

$$\begin{aligned} F(\tau )\le f(\tau ),\quad \tau \ge 0. \end{aligned}$$

(4.6)

Now,

$$\begin{aligned} \left( \frac{f(\tau )}{\tau }\right) ' =\kappa \tau ^{-1}\left( \int _0^\tau \xi ^{p-2}[\log (e+\xi )]^q\,{\textrm{d}}\xi \right) \,{\textrm{d}}s-L\tau ^{-2}>0 \end{aligned}$$

for all $\tau$ large enough ($\tau >R=R(\kappa , L)$). Hence, f satisfies hypothesis (A1) of Theorem 4.1. Furthermore, by (4.4) and (4.5), f also satisfies hypothesis (A2) of Theorem 4.1.

Hence, by Theorem 4.1, there exists $T>0$ and a solution v in $Q_T$ of problem (P) with F replaced by f, with v satisfying

$$\begin{aligned} 0\le v(x,t)\le 2[S(t)\mu ](x)+R\le Ct^{-\frac{N}{\theta }} \end{aligned}$$

in $Q_T$ for some $C>0$. This together with Lemma 2.3 implies that problem (P) possesses a solution u in $Q_T$ such that

$$\begin{aligned} 0\le u(x,t)\le v(x,t)\le 2[S(t)\mu ](x)+R\le Ct^{-\frac{N}{\theta }} \end{aligned}$$

in $Q_T$. Thus, Corollary 4.1 follows. $\square$

4.2 Sufficiency: case (B)

We consider nonlinearities F which generalize case (B).

Theorem 4.2

Let $\mu \in {{\mathcal {L}}}_0$ and let F be an increasing, nonnegative continuous function in $[0,\infty )$. Assume that there exist $R>0$, $\alpha >0$, and positive functions $G\in C([R,\infty ))$ and $H\in C^1([R,\infty ))$ satisfying the following conditions (B1)–(B5):

• (B1) $\tau ^{-p_\theta }F(\tau )\asymp G(\tau )$ as $\tau \rightarrow \infty$;

• (B2) (i) for any $a\ge 1$ and $b>0$, $G(a\tau ^b)\asymp G(\tau )$ as $\tau \rightarrow \infty$. Furthermore, (ii) ${\lim _{\tau \rightarrow \infty }}\tau ^{-\delta }G(\tau )=0$ for all $\delta >0$;

• (B3) (i) $H'(\tau )\asymp \tau ^{-1}G(\tau )>0$ and (ii) $G(\tau H(\tau )^{-1})\asymp G(\tau )$ as $\tau \rightarrow \infty$. Furthermore, (iii) ${\lim _{\tau \rightarrow \infty }}H(\tau )=\infty$ and (iv) ${\lim _{\tau \rightarrow \infty }}\tau ^{-\delta }H(\tau )=0$ for all $\delta >0$;

• (B4) there exists a strictly increasing and convex function $\Phi _\alpha$ in $[R,\infty )$ such that

  $$\begin{aligned} \Phi _\alpha ^{-1}(\tau )=\tau H(\tau )^{-\alpha } \end{aligned}$$

  for all $\tau \in [\Phi _\alpha (R),\infty )$;

• (B5) there exists $\eta \in (0,\theta /N)$ such that the function $P:(R,\infty )\ni \tau \mapsto \tau ^\eta H(\tau )^{-\alpha } G(\tau )$ is increasing.

Then, there exists $\varepsilon >0$ such that if $\mu$ satisfies

(4.7)

for all small enough $\sigma >0$, then problem (P) possesses a solution u in $Q_T$ for some $T>0$, with u satisfying

$$\begin{aligned} 0\le u(x,t)\le \Phi _\alpha ^{-1}[S(t)\Phi _\alpha (\mu +C)] \le Ct^{-\frac{N}{\theta }}H(t^{-1})^{-\frac{N}{\theta }} \end{aligned}$$

in $Q_T$ for some $C>0$.

We prepare a preliminary lemma.

Lemma 4.1

Let $R>0$ and $\alpha >0$. Let G and H be positive functions in $[R,\infty )$ such that $G\in C([R,\infty ))$ and $H\in C^1([R,\infty ))$. Assume also that conditions (B2)-(i), (B3)-(i), (iii), (iv), and (B4) in Theorem 4.2 hold. Then, for any $a>0$, $b>0$, and $c\in {{\mathbb {R}}}$,

$$\begin{aligned} H(a\tau ^b)&\asymp&H(\tau ),\end{aligned}$$

(4.8)

$$\begin{aligned} H(\tau ^b H(\tau )^c)&\asymp&H(\tau ),\end{aligned}$$

(4.9)

$$\begin{aligned} \Phi _\alpha (\tau )&\asymp&\tau H(\tau )^\alpha , \end{aligned}$$

(4.10)

as $\tau \rightarrow \infty$.

Proof

We first prove (4.8). Consider the case where $a\ge 1$ and $b\ge 1$. By (B3)-(i), we see that H is increasing for large enough $\tau$. Then, we take large enough $R'\in (R,\infty )$ so that $a\tau ^b\ge \tau \ge R'$ for $\tau \in [R',\infty )$, $(R'/a)^{1/b}\ge R$, and

$$\begin{aligned} \begin{aligned}&H(\tau )\le H(a\tau ^b) =\int _{R'}^{a\tau ^b} H'(s)\,{\textrm{d}}s+H(R') \le C\int _{R'}^{a\tau ^b} s^{-1}G(s)\,{\textrm{d}}s+H(R')\\&\quad =C\int ^\tau _{(R'/a)^{1/b}} \xi ^{-1}G(a\xi ^b)\,{\textrm{d}}\xi +H(R') \le C\int ^\tau _R \xi ^{-1}G(a\xi ^b)\,{\textrm{d}}\xi +H(\tau ) \end{aligned} \end{aligned}$$

for all $\tau \in [R',\infty )$, where $\xi =(s/a)^{1/b}$. Then, by (B2)-(i) and (B3)-(i), (iii) we take large enough $R''\in (R',\infty )$ so that

$$\begin{aligned}&H(\tau ) \le H(a\tau ^b)\le C\int ^{R''}_R \xi ^{-1}G(a\xi ^b)\,{\textrm{d}}\xi +C\int ^\tau _{R''} \xi ^{-1}G(a\xi ^b)\,{\textrm{d}}\xi +H(\tau )\nonumber \\&\le C+C\int _{R''}^\tau \xi ^{-1}G(\xi )\,{\textrm{d}}\xi +H(\tau ) \le C+C\int _{R''}^\tau H'(\xi )\,{\textrm{d}}\xi \nonumber \\&+H(\tau )\le CH(\tau )+C\le CH(\tau ) \end{aligned}$$

for large enough $\tau$. Thus, (4.8) holds for $a\ge 1$ and $b\ge 1$. In particular, we have

$$\begin{aligned} H(\tau )\asymp H(a\tau )\asymp H(\tau ^b) \end{aligned}$$

(4.11)

as $\tau \rightarrow \infty$ for $a\ge 1$ and $b\ge 1$. Then, we see that

$$\begin{aligned} H(a^{-1}\tau )\asymp H(a\cdot a^{-1}\tau )=H(\tau ), \quad H(\tau ^{1/b})\asymp H((\tau ^{1/b})^b)=H(\tau ), \end{aligned}$$

(4.12)

as $\tau \rightarrow \infty$ for $a\ge 1$ and $b\ge 1$. By (4.11) and (4.12), for any $a>0$ and $b>0$, we obtain

$$\begin{aligned} H(a\tau ^b)\asymp H(\tau ^b)\asymp H(\tau ) \end{aligned}$$

as $\tau \rightarrow \infty$, and (4.8) holds.

Next, we prove (4.9). Let $\delta >0$ be such that $b-\delta |c|>0$. By (B3)-(iii), (iv), we see that $1\le H(\tau )\le \tau ^{\delta }$ for large enough $\tau$. Since H is increasing for large enough $\tau$, we have

$$\begin{aligned} H(\tau ^{b-|c|\delta })\le H(\tau ^b H(\tau )^c)\le H(\tau ^{b+|c|\delta }) \end{aligned}$$

as $\tau \rightarrow \infty$. This together with (4.8) implies that $H(\tau ^b H(\tau )^c)\asymp H(\tau )$ as $\tau \rightarrow \infty$, that is, (4.9) holds.

Furthermore, we observe from (B4) and (4.9) (with $b=1$ and $c=\alpha$) that

$$\begin{aligned} \Phi _\alpha ^{-1}(\tau H(\tau )^\alpha )=\tau H(\tau )^\alpha H(\tau H(\tau )^\alpha )^{-\alpha }\asymp \tau \end{aligned}$$

as $\tau \rightarrow \infty$. Then, we find $C\ge 1$ such that

$$\begin{aligned} C^{-1}\tau \le \Phi _\alpha ^{-1}(\tau H(\tau )^\alpha )\le C\tau \end{aligned}$$

for large enough $\tau$, which together with (B4) implies that

$$\begin{aligned} \Phi _\alpha (C^{-1}\tau )\le \tau H(\tau )^\alpha \le \Phi _\alpha (C\tau ) \end{aligned}$$

for large enough $\tau$. Then, by (4.8) we see that

$$\begin{aligned} \Phi _\alpha (\tau )\le C\tau H(C\tau )^\alpha \le C\tau H(\tau )^\alpha , \qquad \Phi _\alpha (\tau )\ge C^{-1}\tau H(C^{-1}\tau )^\alpha \ge C\tau H(\tau )^\alpha \end{aligned}$$

as $\tau \rightarrow \infty$, yielding (4.10). The proof of Lemma 4.1 is complete. $\square$

Proof of Theorem 4.2

Let $\varepsilon \in (0,1)$ and $L\in (R,\infty )$ be chosen later. Set

$$\begin{aligned} v(x,t):=[S(t)\Phi _\alpha (\mu +L)](x),\quad w(x,t):=2\Phi _\alpha ^{-1}(v(x,t)),\quad \rho (\tau ):=\tau ^{-N}H(\tau ^{-1})^{-\frac{N}{\theta }}. \end{aligned}$$

It follows from (4.8) that

$$\begin{aligned} \rho (t^{\frac{1}{\theta }})= t^{-\frac{N}{\theta }}H\big (t^{-\frac{1}{\theta }}\big )^{-\frac{N}{\theta }} \le Ct^{-\frac{N}{\theta }}H(t^{-1})^{-\frac{N}{\theta }} \end{aligned}$$

(4.13)

for all $t\in (0,T)$ and small enough T. Furthermore, by (B3)-(iv) we see that $\rho (t^{\frac{1}{\theta }})\rightarrow \infty$ as $t\rightarrow 0$. We apply Lemmas 2.1 and 4.1 to obtain

$$\begin{aligned} \Phi _\alpha (L)\le v(x,t)&\le Ct^{-\frac{N}{\theta }}\sup _{z\in {{ \mathbb {R}}}^N}\int _{B(z,t^{\frac{1}{\theta }})}\Phi _\alpha (\mu (y)+L)\,{\textrm{d}}y&\text {[by Lemma~2.1]} \nonumber \\&\le C\Phi _\alpha \left( \varepsilon \rho (t^{\frac{1}{\theta }})\right) \le C\varepsilon \rho (t^{\frac{1}{\theta }})H\left( \varepsilon \rho (t^{\frac{1}{\theta }})\right) ^\alpha&\text {[by (4.7), (4.10)]} \nonumber \\&\le C\varepsilon t^{-\frac{N}{\theta }}H(t^{-1})^{-\frac{N}{\theta }} H\left( C\varepsilon t^{-\frac{N}{\theta }}H(t^{-1})^{-\frac{N}{\theta }}\right) ^\alpha&\text {[by (B3)-(i), (4.13)]} \end{aligned}$$

(4.14)

in $Q_T$ for small enough T. Since

$$\begin{aligned} Ct^{-\frac{N}{\theta }} H(t^{-1})^{-\frac{N}{\theta }+\alpha } \ge C\varepsilon t^{-\frac{N}{\theta }} H(t^{-1})^{-\frac{N}{\theta }+\alpha } \ge t^{-\frac{N}{2\theta }} H(t^{-1})^{-\frac{N}{\theta }+\alpha }\rightarrow \infty \qquad \end{aligned}$$

(4.15)

as $t\rightarrow 0$ (see (B3)-(iii), (iv)), by (B3)-(i), (4.8), (4.9), and (4.14), we have

$$\begin{aligned} \Phi _\alpha (L)\le v(x,t) \le C\varepsilon t^{-\frac{N}{\theta }}H(t^{-1})^{-\frac{N}{\theta }} H\left( Ct^{-\frac{N}{\theta }}H(t^{-1})^{-\frac{N}{\theta }}\right) ^\alpha \le C\varepsilon t^{-\frac{N}{\theta }} H(t^{-1})^{-\frac{N}{\theta }+\alpha } \nonumber \\ \end{aligned}$$

(4.16)

in $Q_T$ for small enough T. By (B4) and (4.16), we have

$$\begin{aligned} 2L\le w(x,t)\le & {} 2\Phi _\alpha ^{-1}\left( C\varepsilon t^{-\frac{N}{\theta }} H(t^{-1})^{-\frac{N}{\theta }+\alpha }\right) \nonumber \\= & {} C\varepsilon t^{-\frac{N}{\theta }} H(t^{-1})^{-\frac{N}{\theta }+\alpha } H\left( C\varepsilon t^{-\frac{N}{\theta }} H(t^{-1})^{-\frac{N}{\theta }+\alpha }\right) ^{-\alpha } \qquad \end{aligned}$$

(4.17)

in $Q_T$ for small enough T. Since $H^{-\alpha }$ is monotone decreasing for large enough $\tau$, by (4.15) we have

$$\begin{aligned} H\left( C\varepsilon t^{-\frac{N}{\theta }} H(t^{-1})^{-\frac{N}{\theta }+\alpha }\right) ^{-\alpha } \le H\left( t^{-\frac{N}{2\theta }} H(t^{-1})^{-\frac{N}{\theta }+\alpha }\right) ^{-\alpha } \end{aligned}$$

for all $t\in (0,T)$ and small enough T. This together with (4.9) implies that

$$\begin{aligned} H\left( C\varepsilon t^{-\frac{N}{\theta }} H(t^{-1})^{-\frac{N}{\theta }+\alpha }\right) ^{-\alpha } \le CH(t^{-1})^{-\alpha } \end{aligned}$$

(4.18)

for all $t\in (0,T)$ and small enough T. By (4.17) and (4.18), we obtain

$$\begin{aligned} 2L\le w(x,t) \le C\varepsilon t^{-\frac{N}{\theta }}H(t^{-1})^{-\frac{N}{\theta }} \end{aligned}$$

(4.19)

in $Q_T$ for small enough T. Then, taking large enough L if necessary, by (B1) and (4.10) we have

$$\begin{aligned} \frac{F(w(x,t))}{v(x,s)}=\frac{F(w(x,t))}{\Phi _\alpha (w(x,t)/2)} \le C\frac{w(x,t)^{p_\theta }G(w(x,t))}{w(x,t)H(w(x,t))^\alpha } =Cw(x,t)^{\frac{\theta }{N}-\eta }P(w(x,t))\nonumber \\ \end{aligned}$$

(4.20)

in $Q_T$, where P is as in (B5). Furthermore, by (B5) and (4.19) we obtain

$$\begin{aligned}{} & {} P(w(x,t))\le P\left( C\varepsilon t^{-\frac{N}{\theta }}H(t^{-1})^{-\frac{N}{\theta }}\right) \le P\left( Ct^{-\frac{N}{\theta }}H(t^{-1})^{-\frac{N}{\theta }}\right) \nonumber \\{} & {} \quad =\left( Ct^{-\frac{N}{\theta }}H(t^{-1})^{-\frac{N}{\theta }}\right) ^\eta H\left( Ct^{-\frac{N}{\theta }}H(t^{-1})^{-\frac{N}{\theta }}\right) ^{-\alpha } G\left( Ct^{-\frac{N}{\theta }}H(t^{-1})^{-\frac{N}{\theta }}\right) \qquad \qquad \end{aligned}$$

(4.21)

in $Q_T$ for small enough T. On the other hand, by (B3)-(iv) we see that $t^{-1}H(t^{-1})^{-1}\rightarrow \infty$ as $t\rightarrow 0$. Then, by (B2)-(i) and (B3)-(ii) we see that

$$\begin{aligned} G\left( Ct^{-\frac{N}{\theta }}H(t^{-1})^{-\frac{N}{\theta }}\right) \le CG\left( t^{-1}H(t^{-1})^{-1}\right) \le CG(t^{-1}) \end{aligned}$$

for all $t\in (0,T)$ and small enough T. This together with (4.8), (4.9), and (4.21) implies that

$$\begin{aligned} P(w(x,t))\le C\left( t^{-\frac{N}{\theta }}H(t^{-1})^{-\frac{N}{\theta }}\right) ^\eta H(t^{-1})^{-\alpha }G(t^{-1}) \end{aligned}$$

(4.22)

in $Q_T$. Since $0<\eta <\theta /N$ (see (B5)), by (4.19), (4.20), and (4.22) we obtain

$$\begin{aligned} \frac{F(w(x,t))}{v(x,s)}\le & {} C\left( C\varepsilon t^{-\frac{N}{\theta }}H(t^{-1})^{-\frac{N}{\theta }}\right) ^{\frac{\theta }{N}-\eta } \left( t^{-\frac{N}{\theta }}H(t^{-1})^{-\frac{N}{\theta }}\right) ^\eta H(t^{-1})^{-\alpha }G(t^{-1})\nonumber \\\le & {} C\varepsilon ^{\frac{\theta }{N}-\eta } t^{-1}H(t^{-1})^{-1-\alpha }G(t^{-1}) \end{aligned}$$

(4.23)

in $Q_T$. Therefore, we deduce from (B3)-(i) and (4.23) that

$$\begin{aligned} \begin{aligned} \int _0^t \left\| \frac{F(w(s))}{v(s)}\right\| _\infty \,{\textrm{d}}s&\le C \varepsilon ^{\frac{\theta }{N}-\eta }\int _0^t s^{-1}H(s^{-1})^{-1-\alpha }G(s^{-1})\,{\textrm{d}}s\\&\le C \varepsilon ^{\frac{\theta }{N}-\eta } \int _{t^{-1}}^\infty \tau ^{-1}H(\tau )^{-1-\alpha }G(\tau )\,{\textrm{d}}\tau \\&\le C \varepsilon ^{\frac{\theta }{N}-\eta } \int _{t^{-1}}^\infty H(\tau )^{-1-\alpha }H'(\tau )\,{\textrm{d}}\tau \\&\le C \varepsilon ^{\frac{\theta }{N}-\eta }H(t^{-1})^{-\alpha } \end{aligned} \end{aligned}$$

(4.24)

for all $t\in (0,T)$. Similarly, by (4.14) and Lemma 4.1, we have

$$\begin{aligned} \begin{aligned}&\frac{[S(t)\Phi _\alpha (\mu +L)](x)}{w(x,t)} =\frac{v(x,t)}{2\Phi _\alpha ^{-1}(v(x,t))}=\frac{1}{2}H(v(x,t))^\alpha \le CH(t^{-1})^\alpha \end{aligned} \end{aligned}$$

(4.25)

in $Q_T$. Therefore, taking small enough $\varepsilon \in (0,1)$, by (4.24) and (4.25) we obtain

$$\begin{aligned} \begin{aligned}&[S(t)\mu ](x)+\int _0^t S(t-s)F(w(s))\,{\textrm{d}}s\\&\le \frac{1}{2}w(x,t)+\int _0^t \left\| \frac{F(w(s))}{v(s)}\right\| _\infty S(t-s)v(s)\,{\textrm{d}}s\\&=\frac{1}{2}w(x,t)+w(x,t)\frac{[S(t)\Phi _\alpha (\mu +L)](x)}{w(x,t)}\int _0^t \left\| \frac{F(w(s))}{v(s)}\right\| _\infty \,{\textrm{d}}s\\&\le w(x,t)\left[ \frac{1}{2}+C\varepsilon ^{\frac{\theta }{N}-\eta }\right] \le w(x,t) \end{aligned} \end{aligned}$$

in $Q_T$, where we have used the fact that

$$\begin{aligned} w(x,t)\ge 2\Phi ^{-1}_\alpha \left( S(t)\Phi _\alpha (\mu )\right) \ge 2S(t)\mu \end{aligned}$$

by Jensen’s inequality. Hence, w is a supersolution in $Q_T$ and Theorem 4.2 now follows from Lemma 2.2 and (4.19). $\square$

Corollary 4.2

Let $\mu \in {{\mathcal {L}}}_0$ and assume conditions (F1) and (F2) hold with $p=p_\theta$ and $q\ge -1$. Let $\alpha >0$ and set

$$\begin{aligned} \begin{array}{ll} h(\tau ) &{} := \left\{ \begin{array}{ll} (\log (e+\tau ))^{q+1} &{}\quad \hbox {if}\quad q>-1,\\ \log (e+\log (e+\tau )) &{}\quad \hbox {if}\quad q=-1, \end{array} \right. \\ \psi _\alpha ^\pm (\tau ) &{} :=\tau h(\tau )^{\pm \alpha }, \end{array} \end{aligned}$$

for $\tau \in (0,\infty )$. Then, there exists $\varepsilon >0$ such that if $\mu$ satisfies

(4.26)

for small enough $\sigma >0$, then problem (P) possesses a solution u in $Q_T$ for some $T>0$, with u satisfying

$$\begin{aligned} 0\le u(x,t)\le \left\{ \begin{array}{ll} Ct^{-\frac{N}{\theta }}|\log t|^{-\frac{N(q+1)}{\theta }} &{}\quad \hbox {if}\quad q>-1,\\ Ct^{-\frac{N}{\theta }}[\log |\log t|]^{-\frac{N}{\theta }} &{}\quad \hbox {if}\quad q=-1 \end{array} \right. \end{aligned}$$

in $Q_T$ for some $C>0$.

Proof

Let $\alpha >1$. Set

$$\begin{aligned} G(\tau ):=(\log \tau )^q,\quad H(\tau ):= \left\{ \begin{array}{ll} (\log \tau )^{q+1} &{}\quad \hbox {if}\quad q>-1,\\ \log (\log \tau ) &{}\quad \hbox {if}\quad q=-1, \end{array} \right. \quad \Psi _\alpha (\tau ):=\tau H(\tau )^{-\alpha }. \end{aligned}$$

Then, for any $a\ge 1$ and $b>0$, we have

$$\begin{aligned} G(a\tau ^b)&=(\log a+b\log \tau )^q=b^q(\log \tau )^q(1+o(1))\\&\asymp (\log \tau )^q=G(\tau ) \quad \hbox {for any } a\ge 1 \hbox { and }b>0,\\ G(\tau )&=o(\tau ^\delta )\quad \hbox {for any } \delta >0, \end{aligned}$$

as $\tau \rightarrow \infty$. Thus, condition (B2) holds. Let $\Phi _\alpha$ be the inverse function of $\Psi _\alpha$, and we show that conditions (B3)–(B5) in Theorem 4.2 hold.

Consider the case of $q>-1$. Then,

$$\begin{aligned} \begin{aligned} H'(\tau )&=(q+1)\tau ^{-1}(\log \tau )^q,\\ \Psi _\alpha '(\tau )&=(\log \tau )^{-\alpha (q+1)}-\alpha (q+1)(\log \tau )^{-\alpha (q+1)-1}=(\log \tau )^{-\alpha (q+1)}(1+o(1))>0,\\ \Psi _\alpha ''(\tau )&=-\alpha (q+1)\tau ^{-1}(\log \tau )^{-\alpha (q+1)-1}+\alpha (q+1)(\alpha (q+1)+1)\tau ^{-1}(\log \tau )^{-\alpha (q+1)-2}\\&=-\alpha (q+1)\tau ^{-1}(\log \tau )^{-\alpha (q+1)-1}(1+o(1))<0, \end{aligned} \end{aligned}$$

as $\tau \rightarrow \infty$. We see that

$$\begin{aligned}&\tau ^{-1}G(\tau )=\tau ^{-1}(\log \tau )^q\asymp H'(\tau )\quad \hbox { as }\tau \rightarrow \infty ,\\&G(\tau H(\tau )^{-1})=\left[ \log (\tau (\log \tau )^{-(q+1)})\right] ^q\asymp (\log \tau )^q=G(\tau )\quad \hbox {as } \tau \rightarrow \infty ,\\&\lim _{\tau \rightarrow \infty }H(\tau )=\infty ,\qquad H(\tau )=o(\tau ^\delta )\quad \hbox {as } \tau \rightarrow \infty \hbox { for any } \delta >0. \end{aligned}$$

Thus, condition (B3) holds. Furthermore, we observe that $\Psi _\alpha$ is strictly increasing and concave for large enough $\tau$, that is, the inverse function $\Phi _\alpha$ of $\Psi _\alpha ^{-1}$ exists and it is strictly increasing and convex for large enough $\tau$. Then, condition (B4) holds. In addition, for any $\eta >0$, setting

$$\begin{aligned} P(\tau )=\tau ^\eta H(\tau )^{-\alpha }G(\tau )=\tau ^\eta (\log \tau )^{-\alpha (q+1)+q}, \end{aligned}$$

by Lemma 2.6, we see that $P'(\tau )>0$ for large enough $\tau$. This implies that condition (B5) also holds. Thus, conditions (B3)–(B5) hold in the case of $q>-1$.

Consider the case of $q=-1$. It follows that

$$\begin{aligned} \begin{aligned} H'(\tau )&=\tau ^{-1}(\log \tau )^{-1},\\ \Psi _\alpha '(\tau )&=(\log (\log \tau ))^{-\alpha }-\alpha (\log \tau )^{-1}(\log (\log \tau ))^{-\alpha -1} =(\log (\log \tau ))^{-\alpha }(1+o(1))>0,\\ \Psi _\alpha ''(\tau )&=-\alpha \tau ^{-1}(\log \tau )^{-1}(\log (\log \tau ))^{-\alpha -1} +\alpha \tau ^{-1}(\log \tau )^{-2}(\log (\log \tau ))^{-\alpha -1}\\&\quad +\alpha (\alpha +1)\tau ^{-1}(\log \tau )^{-2}(\log (\log \tau ))^{-\alpha -2}\\&=-\alpha \tau ^{-1}(\log \tau )^{-1}(\log (\log \tau ))^{-\alpha -1}(1+o(1))<0, \end{aligned} \end{aligned}$$

as $\tau \rightarrow \infty$. Similarly to the case of $q>-1$, we have

$$\begin{aligned}&\tau ^{-1}G(\tau )=\tau ^{-1}(\log \tau )^{-1}=H'(\tau )\quad \hbox {as } \tau \rightarrow \infty ,\\&G(\tau H(\tau )^{-1})=\left[ \log (\tau (\log (\log \tau ))^{-1})\right] ^{-1}\asymp (\log \tau )^{-1}=G(\tau )\quad \hbox {as }\tau \rightarrow \infty ,\\&\lim _{\tau \rightarrow \infty }H(\tau )=\infty ,\qquad H(\tau )=o(\tau ^\delta )\quad \hbox {as } \tau \rightarrow \infty \hbox { for any }\delta >0. \end{aligned}$$

Thus, condition (B3) holds. Furthermore, we see that $\Psi _\alpha$ is strictly increasing and concave for large enough $\tau$, that is, the inverse function $\Psi _\alpha ^{-1}$ exists and it is strictly increasing and convex for large enough $\tau$. Then, condition (B4) holds. In addition, for any $\eta >0$, setting

$$\begin{aligned} P(\tau )=\tau ^\eta H(\tau )^{-\alpha }G(\tau )=\tau ^\eta (\log (\log \tau ))^{-\alpha }(\log \tau )^{-1}, \end{aligned}$$

by Lemma 2.6 we see that $P'(\tau )>0$ for large enough $\tau$. This implies that condition (B5) also holds. Thus, conditions (B3)–(B5) hold in the case of $q=-1$.

Assume (4.26). By Lemma 2.6 (with $a=1$, $b=-\alpha (q+1)$, and $c=0$ for $q>-1$ and with $a=1$, $b=0$, and $c=-\alpha$ for $q=-1$), we have

$$\begin{aligned} \Phi _\alpha (\tau )=\Psi _\alpha ^{-1}(\tau )\asymp \left\{ \begin{array}{ll} \tau (\log \tau )^{\alpha (q+1)} &{}\quad \hbox {for}\quad q>-1,\\ \tau (\log (\log \tau ))^\alpha &{}\quad \hbox {for}\quad q=-1, \end{array} \right. \end{aligned}$$

as $\tau \rightarrow \infty$. Since

$$\begin{aligned} \Phi _\alpha ^{-1}(\tau )=\Psi _\alpha (\tau )\le C\psi _\alpha ^-(\tau ), \qquad \Phi _\alpha (\tau )=\Psi _\alpha ^{-1}(\tau )\le C\psi _\alpha ^{+}(\tau ), \end{aligned}$$

for large enough $\tau$, taking large enough $R>0$ if necessary, we see that

(4.27)

Furthermore, we see that

$$\begin{aligned} \psi _\alpha ^+(\tau +R)\le C\psi _\alpha ^+(\tau )+C, \quad \psi _\alpha ^-(C\tau +C)\le C\psi _\alpha ^-(\tau )+C \end{aligned}$$

for $\tau >0$. Then, by (4.26) and (4.27) we see that

for all small enough $\sigma >0$.

Let f be as in (4.5). Since $f(\tau )\asymp \tau ^{p_\theta }(\log \tau )^q$ as $\tau \rightarrow \infty$, condition (B1) holds with F replaced by f. We deduce from Theorem 4.2 that problem (P) with F replaced by f possesses a solution v in $Q_T$ for some $T>0$ such that

$$\begin{aligned} 0\le v(x,t)\le \left\{ \begin{array}{ll} Ct^{-\frac{N}{\theta }}|\log t|^{-\frac{N(q+1)}{\theta }} &{}\quad \hbox {if}\quad q>-1,\\ Ct^{-\frac{N}{\theta }}[\log |\log t|]^{-\frac{N}{\theta }} &{}\quad \hbox {if}\quad q=-1, \end{array} \right. \end{aligned}$$

for all $(x,t)\in Q_T$. This together with $f(\tau )\ge F(\tau )$ (by (4.6)) and Lemma 2.3 implies that problem (P) possesses a solution u in $Q_T$ such that

$$\begin{aligned} 0\le u(x,t)\le v(x,t)\le \left\{ \begin{array}{ll} Ct^{-\frac{N}{\theta }}|\log t|^{-\frac{N(q+1)}{\theta }} &{}\quad \hbox {if}\quad q>-1,\\ Ct^{-\frac{N}{\theta }}[\log |\log t|]^{-\frac{N}{\theta }} &{}\quad \hbox {if}\quad q=-1, \end{array} \right. \end{aligned}$$

for all $(x,t)\in Q_T$. Thus, Corollary 4.2 follows. $\square$

4.3 Sufficiency: case (C)

In this section, we consider nonlinearities F which generalize case (C).

Theorem 4.3

Let $\mu \in {{\mathcal {L}}}_0$ and let F be an increasing, nonnegative continuous function in $[0,\infty )$ such that

• (C1) there exist $R\ge 0$ and $d>1$ such that the function $(R,\infty )\ni \tau \mapsto \tau ^{-d}F(\tau )\in (0,\infty )$ is increasing.

Furthermore, assume that there exists a continuous function G in $[R,\infty )$ satisfying the following conditions:

• (C2) there exists $p\in [d,d+1)$ such that $G(\tau )\succeq \tau ^{-p}F(\tau )>0$ as $\tau \rightarrow \infty$;

• (C3) for any $a\ge 1$, $b>0$, and $c\in {{ \mathbb {R}}}$, $G(a\tau ^b G(\tau )^c)\asymp G(\tau )$ as $\tau \rightarrow \infty$;

• (C4) there exists $\delta \in (0,1)$ such that the function $(R,\infty )\ni \tau \mapsto \tau ^{-\delta }G(\tau )$ is decreasing.

Let $\alpha >1$. Then, there exists $\varepsilon >0$ such that if $\mu$ satisfies

(4.28)

for small enough $\sigma >0$, then problem (P) possesses a solution u in $Q_T$ for some $T>0$, with u satisfying

$$\begin{aligned} 0\le u(x,t)\le 2[S(t)\mu ^\alpha ](x)^{\frac{1}{\alpha }}+R\le Ct^{-\frac{1}{p-1}} G(t^{-1})^{-\frac{1}{p-1}} \end{aligned}$$

in $Q_T$ for some $C>0$.

Proof

Let $\varepsilon \in (0,1)$ be chosen later, and assume (4.28). Without loss of generality, we may assume that $\alpha \in (1,d)$. Indeed, if $\alpha \ge d$ and (4.28) holds, then for any $\alpha '\in (1,d)$ we can write $\mu ^{\alpha }=(\mu ^{\alpha '})^\frac{\alpha }{\alpha '}$ and apply Jensen’s inequality to give

for small enough $\sigma >0$. Consequently, (4.28) also holds for $\alpha '\in (1,d)$.

Set

$$\begin{aligned} w(x,t):=2[S(t)\mu ^\alpha ](x)^{\frac{1}{\alpha }}+R. \end{aligned}$$

(4.29)

It follows from (C3), Lemma 2.1, and (4.28) that

$$\begin{aligned} \begin{aligned} 0&\le [S(t)\mu ^\alpha ](x) \le Ct^{-\frac{N}{\theta }}\sup _{z\in {{ \mathbb {R}}}^N}\int _{B(z,t^{\frac{1}{\theta }})}\mu (y)^\alpha \,{\textrm{d}}y\\&\le C\varepsilon ^\alpha t^{-\frac{\alpha }{p-1}}G(t^{-\frac{1}{\theta }})^{-\frac{\alpha }{p-1}} \le C\varepsilon ^\alpha t^{-\frac{\alpha }{p-1}}G(t^{-1})^{-\frac{\alpha }{p-1}} \end{aligned} \end{aligned}$$

in $Q_T$ for small enough T. On the other hand, by (C1) and (C2) we see that

$$\begin{aligned} \lim _{\tau \rightarrow \infty }\tau G(\tau )\ge C\lim _{\tau \rightarrow \infty }\tau ^{1-p}F(\tau )=C\lim _{\tau \rightarrow \infty }\tau ^{d+1-p}\tau ^{-d}F(\tau )=\infty , \end{aligned}$$

since $p<d+1$. These imply that

$$\begin{aligned} R\le w(x,t) \le R+C \varepsilon t^{-\frac{1}{p-1}} G(t^{-1})^{-\frac{1}{p-1}}&\le C \varepsilon t^{-\frac{1}{p-1}} G(t^{-1})^{-\frac{1}{p-1}}\end{aligned}$$

(4.30)

$$\begin{aligned}&\le Ct^{-\frac{1}{p-1}} G(t^{-1})^{-\frac{1}{p-1}} \end{aligned}$$

(4.31)

in $Q_T$. Since $1<\alpha <d$, by (C1)–(C3), (4.30), and (4.31) we obtain

$$\begin{aligned}{} & {} \frac{F(w(x,t))}{w(x,t)^\alpha } =w(x,t)^{{d-\alpha }}\frac{F(w(x,t))}{w(x,t)^d} \nonumber \\{} & {} \quad \le C\left[ C\varepsilon t^{-\frac{1}{p-1}}G(t^{-1})^{-\frac{1}{p-1}}\right] ^{{d-\alpha }} \left[ Ct^{-\frac{1}{p-1}}G(t^{-1})^{-\frac{1}{p-1}}\right] ^{-d} F\left( Ct^{-\frac{1}{p-1}}G(t^{-1})^{-\frac{1}{p-1}}\right) \nonumber \\{} & {} \quad \le C\left[ C\varepsilon t^{-\frac{1}{p-1}}G(t^{-1})^{-\frac{1}{p-1}}\right] ^{{d-\alpha }} \left[ Ct^{-\frac{1}{p-1}}G(t^{-1})^{-\frac{1}{p-1}}\right] ^{p-d} G\left( Ct^{-\frac{1}{p-1}}G(t^{-1})^{-\frac{1}{p-1}}\right) \nonumber \\{} & {} \quad \le C\varepsilon ^{d-\alpha }t^{-\frac{p-\alpha }{p-1}}G(t^{-1})^{\frac{\alpha -1}{p-1}} \end{aligned}$$

(4.32)

in $Q_T$ for small enough T. Similarly, by (4.31) we have

$$\begin{aligned} \begin{aligned} {w(x,t)^{\alpha -1}} \le Ct^{-\frac{\alpha -1}{p-1}}G(t^{-1})^{-\frac{\alpha -1}{p-1}} \end{aligned} \end{aligned}$$

(4.33)

in $Q_T$. On the other hand, by (C4) we see that

$$\begin{aligned} \begin{aligned}&\int _0^t s^{-\frac{p-\alpha }{p-1}}G(s^{-1})^{\frac{\alpha -1}{p-1}}\,{\textrm{d}}s =\int _{t^{-1}}^\infty \tau ^{\frac{p-\alpha }{p-1}-2}G(\tau )^{\frac{\alpha -1}{p-1}}\,{\textrm{d}}\tau \\&=\int _{t^{-1}}^\infty \tau ^{-1-(1-\delta )\frac{\alpha -1}{p-1}} [\tau ^{-\delta }G(\tau )]^{\frac{\alpha -1}{p-1}}\,{\textrm{d}}\tau \\&\le C[t^\delta G(t^{-1})]^{\frac{\alpha -1}{p-1}}t^{(1-\delta )\frac{\alpha -1}{p-1}} =Ct^{\frac{\alpha -1}{p-1}}G(t^{-1})^{\frac{\alpha -1}{p-1}} \end{aligned} \end{aligned}$$

(4.34)

for all $t\in (0,T)$ and small enough T. Therefore, taking small enough $\varepsilon$, by Jensen’s inequalities, (4.32), (4.33), and (4.34) we obtain

$$\begin{aligned} \begin{aligned}&[S(t)\mu ](x)+\int _0^t S(t-s)F(w(s))\,{\textrm{d}}s\\&\le [S(t)\mu ^\alpha ](x)^{\frac{1}{\alpha }}+C\int _0^t \left\| \frac{F(w(s))}{w(s)^\alpha }\right\| _{L^\infty ({{ \mathbb {R}}}^N)} S(t-s)[S(s)\mu ^\alpha +R^\alpha ]\,{\textrm{d}}s\\&\le \frac{1}{2}w(x,t)+C\varepsilon ^{d-\alpha }[S(t)\mu ^\alpha +R^\alpha ] \int _0^t s^{-\frac{p-\alpha }{p-1}}G(s^{-1})^{\frac{\alpha -1}{p-1}}\,{\textrm{d}}s\\&\le \frac{1}{2}w(x,t)+C\varepsilon ^{d-\alpha }w(x,t)^\alpha \int _0^t s^{-\frac{p-\alpha }{p-1}}G(s^{-1})^{\frac{\alpha -1}{p-1}}\,{\textrm{d}}s\\&\le \frac{1}{2}w(x,t) +C\varepsilon ^{d-\alpha }\left\| w(t)^{\alpha -1}\right\| _{L^\infty ({{ \mathbb {R}}}^N)} t^{\frac{\alpha -1}{p-1}}G(t^{-1})^{\frac{\alpha -1}{p-1}} w(x,t)\\&\le w(x,t)\left[ \frac{1}{2}+C\varepsilon ^{d-\alpha }\right] \le w(x,t) \end{aligned} \end{aligned}$$

in $Q_T$. Hence, w is a supersolution in $Q_T$ and Theorem 4.3 now follows from Lemma 2.2, (4.29), and (4.30). $\square$

Corollary 4.3

Let $\mu \in {{\mathcal {L}}}_0$ and assume conditions (F1) and (F2) hold. For any $\alpha >1$, there exists $\varepsilon >0$ such that if $\mu$ satisfies

(4.35)

for all small enough $\sigma >0$, then problem (P) possesses a solution u in $Q_T$ for some $T>0$, with u satisfying

$$\begin{aligned} 0\le u(x,t)\le 2[S(t)\mu ^\alpha ](x)^{\frac{1}{\alpha }}+R\le Ct^{-\frac{1}{p-1}}|\log t|^{-\frac{q}{p-1}} \end{aligned}$$

in $Q_T$ for some $R, C>0$.

Proof

Let $d\in (1,p)$ with $d>p-1$. Let $\kappa$, $L>0$, and set

$$\begin{aligned} f(\tau ):=\kappa \tau ^d\int _0^\tau s^{-d}\left( \int _0^s \xi ^{p-2}[\log (e+\xi )]^q\,{\textrm{d}}\xi \right) \,{\textrm{d}}s+L \end{aligned}$$

(4.36)

for $\tau \in (0,\infty )$. It follows from Lemma 2.8 (iii) that

$$\begin{aligned} \tau ^d\int _0^\tau s^{-d}\left( \int _0^s \xi ^{p-2}[\log (e+\xi )]^q\,{\textrm{d}}\xi \right) \,{\textrm{d}}s\asymp \tau ^p[\log \tau ]^q \end{aligned}$$

(4.37)

as $\tau \rightarrow \infty$. We take large enough $\kappa$ and L so that $F(\tau )\le f(\tau )$ in $[0,\infty )$. On the other hand, since

$$\begin{aligned} \left( \frac{f(\tau )}{\tau ^d}\right) '=\tau ^{-d} \left[ \kappa \left( \int _0^\tau \xi ^{p-2}[\log (e+\xi )]^q\,{\text {d}}\xi \right) \,{\text {d}}s -L\tau ^{-1}d\right] >0 \end{aligned}$$

for large enough $\tau$, condition (C1) in Theorem 4.3 holds in $(R,\infty )$ with F replaced by f for some $R>0$.

Taking large enough R if necessary and setting $G(\tau ):=(\log \tau )^q$ for $\tau \in (R,\infty )$, we see that the function $(R,\infty )\ni \tau \mapsto \tau ^{-\frac{1}{2}}G(\tau )$ is decreasing (i.e., $\delta =1/2$ in (C4)). By (4.36) and (4.37), we find $C>0$ such that

$$\begin{aligned} \tau ^{-p}f(\tau )\le CG(\tau ) \end{aligned}$$

for all $\tau \in (R,\infty )$. Then, conditions (C2)–(C4) in Theorem 4.3 hold with F replaced by f. Therefore, by Theorem 4.3 there exists $\varepsilon >0$ such that if $\mu$ satisfies (4.35), then problem (P) with F replaced by f possesses a solution v in $Q_T$ for some $T>0$ such that

$$\begin{aligned} 0\le v(x,t)\le 2[S(t)\mu ^\alpha ](x)^{\frac{1}{\alpha }}+R\le Ct^{-\frac{1}{p-1}}|\log t|^{-\frac{q}{p-1}} \end{aligned}$$

in $Q_T$, for some $C>0$. This together with Lemma 2.3 implies that problem (P) possesses a solution u in $Q_T$ such that

$$\begin{aligned} 0\le u(x,t)\le v(x,t)\le 2[S(t)\mu ^\alpha ](x)^{\frac{1}{\alpha }}+R\le Ct^{-\frac{1}{p-1}}|\log t|^{-\frac{q}{p-1}} \end{aligned}$$

in $Q_T$. Thus, Corollary 4.3 follows. $\square$

4.4 A special case: Dirac measure as initial data

Here, we provide a necessary and sufficient condition on the nonlinearity F for the solvability of problem (P) in the special case when $\mu =\delta _y$, the Dirac measure in ${{ \mathbb {R}}}^N$ based at point y. This problem was considered in [7] for the opposite sign pure power law case $F(u)=-u^p$, i.e., dissipative F.

Corollary 4.4

Suppose F satisfies

• (D1) F is nonnegative and locally Lipschitz continuous in $[0,\infty )$;

• (D2) there exist $R>0$ and $d>1$ such that

  • (i) the function $(R,\infty )\ni \tau \mapsto \tau ^{-d}F(\tau )\in (0,\infty )$ is increasing;

  • (ii) F is convex in $(R,\infty )$.

Let $y\in \mathbb {R}^N$. Then, problem (P) possesses a local-in-time solution with $\mu =\delta _y$ if and only if

$$\begin{aligned} \int ^\infty _1 \tau ^{-p_\theta -1}F(\tau )\,{\textrm{d}}\tau <\infty . \end{aligned}$$

(4.38)

Proof

Assume that problem (P) possesses a solution with $\mu =\delta _y$ in $Q_T$ for some $T>0$. Set

$$\begin{aligned} f(\tau ):=0\quad \hbox {for}\quad 0\le \tau \le R, \quad f(\tau ):=F(\tau )-\tau ^dR^{-d}F(R)\quad \hbox {for}\quad \tau >R. \end{aligned}$$

(4.39)

Then, by (D2)-(i) we see that f is increasing and $F\ge f$ in $[0,\infty )$. Applying Theorem 3.1 with $z=y$, so that $m_\sigma (z)=\delta _y(B(y,\sigma ))\equiv 1$, we find $\gamma \ge 1$ such that

$$\begin{aligned} \int _{\gamma ^{-1}T^{-\frac{N}{\theta }}}^{\gamma ^{-1}\sigma ^{-N}} s^{-p_\theta -1}f(s)\,{\textrm{d}}s\le \gamma ^{p_\theta +1}, \qquad 0<\sigma <T^{\frac{1}{\theta }}. \end{aligned}$$

Letting $\sigma \rightarrow 0$, we have

$$\begin{aligned} \int _{\gamma ^{-1}T^{-\frac{N}{\theta }}}^\infty s^{-p_\theta -1}f(s)\,{\textrm{d}}s\le \gamma ^{p_\theta +1}. \end{aligned}$$

This together with (4.39) implies (4.38).

Conversely, under condition (4.38), we apply Theorem 4.1 to obtain a local-in-time solution of problem (P) with $\mu =\delta _y$. Thus, Corollary 4.4 follows. $\square$

We mention that the integral condition (4.38) also appears in [30, Theorem 5.1] as a necessary and sufficient condition for existence with $L^1$ initial data. See also the informal argument preceding the proof of Theorem 4.1 of that work, where a Dirac delta function is considered as initial data.

5 Proof of the main theorem

Proof of Theorem 1.1

Assertion (i) is proved by Corollary 3.1 (ii), Remark 2.1, and Corollary 4.1.

We now prove the nonexistence parts of statements (1) and (2) in assertion (ii). Suppose first that (1.3) holds and there exists a local solution of problem (P). Then,

$$\begin{aligned} \sup _{z\in {{ \mathbb {R}}}^N}\mu (B(z,\sigma )) \ge \gamma _1\int _{B(0,\sigma )}|x|^{-N}|\log |x||^{-1}[\log |\log |x||]^{-\frac{N}{\theta }-1}\,{\textrm{d}}x \ge C_1\gamma _1[\log |\log \sigma |]^{-\frac{N}{\theta }} \end{aligned}$$

for small enough $\sigma >0$. For large enough $\gamma _1$, we then obtain a contradiction to Corollary 3.1. Hence, no local solution can exist for such $\gamma _1$. Now suppose that (1.5) holds. Then, there exists $C_2>0$ such that

$$\begin{aligned} \sup _{z\in {{ \mathbb {R}}}^N}\mu (B(z,\sigma )) \ge \gamma _2\int _{B(0,\sigma )}|x|^{-N}|\log |x||^{-\frac{N(q+1)}{\theta }-1}\,{\textrm{d}}x \ge C_2\gamma _2|\log \sigma |^{-\frac{N(q+1)}{\theta }} \end{aligned}$$

for small enough $\sigma >0$. Again, we can obtain a contradiction to Corollary 3.1 for large enough $\gamma _2$ and deduce that problem (P) possesses no local-in-time solution for such $\gamma _2$.

Next, we prove the existence parts of statements (1) and (2) in assertion (ii). Assume therefore that either (1.4) with $\varepsilon _1\in (0,1)$ or (1.6) with $\varepsilon _2\in (0,1)$ hold. Let $\alpha >0$ and set

$$\begin{aligned} h(\tau ):= \left\{ \begin{array}{ll} \log (e+\log (e+\tau )) &{}\quad \hbox {if (1.4) holds},\\ (\log (e+\tau ))^{q+1} &{}\quad \hbox {if (1.6) holds}, \end{array} \right. \qquad \psi _\alpha ^\pm (\tau ):=\tau h(\tau )^{\pm \alpha }. \end{aligned}$$

If (1.4) holds, then

$$\begin{aligned} \begin{aligned} \psi _\alpha ^+(\mu (x))&\le C\mu (x)\log (e+\log (e+\mu (x)))^\alpha \\&\le C\varepsilon _1|x|^{-N}|\log |x||^{-1}[\log |\log |x||]^{-\frac{N}{\theta }-1+\alpha }\chi _{B(0,R)}(x)+C, \quad x\in { \mathbb {R}}^N. \end{aligned} \end{aligned}$$

This implies that

for small enough $\sigma >0$.

Similarly, if (1.6) holds, then

$$\begin{aligned} \begin{aligned} \psi _\alpha ^+(\mu (x))&\le C\mu (x)[\log (e+\mu (x))]^{\alpha (q+1)}\\&\le C\varepsilon _2|x|^{-N}|\log |x||^{-\frac{N(q+1)}{\theta }-1+\alpha (q+1)}\chi _{B(0,R)}(x)+C, \quad x\in { \mathbb {R}}^N. \end{aligned} \end{aligned}$$

This implies that

for small enough $\sigma >0$. Therefore, by Corollary 4.2 we see that, if $\varepsilon _1>0$ (respectively $\varepsilon _2>0$) is small enough, then problem (P) possesses a local-in-time solution. Thus, statements (1) and (2) in assertion (ii) follow.

Finally, we prove statement (3) in assertion (ii). Assume that (1.7) holds. Then,

$$\begin{aligned} \sup _{z\in {{ \mathbb {R}}}^N}\mu (B(z,\sigma )) \ge \gamma _3\int _{B(0,\sigma )}|x|^{-\frac{\theta }{p-1}}|\log |x||^{-\frac{q}{p-1}}\,{\textrm{d}}x \ge C_1\gamma _3\sigma ^{N-\frac{\theta }{p-1}}|\log \sigma |^{-\frac{q}{p-1}} \end{aligned}$$

for small enough $\sigma >0$. This together with Corollary 3.1 implies that problem (P) possesses no local-in-time solution for large enough $\gamma _3$. Conversely, suppose that (1.8) holds. Since $p>p_\theta$, we find $\alpha >1$ such that $\alpha \theta /(p-1)<N$. Then, we have

for small enough $\sigma >0$. By Corollary 4.3, we see that, if $\varepsilon _3>0$ is small enough, then problem (P) possesses a local-in-time solution. Thus, statement (3) in assertion (ii) follows. The proof is complete. $\square$

Remark 5.1

The arguments in the proof of Theorem 1.1 are readily adapted to further log-refinements. For example, suppose that (F2) is replaced by

(F2$^\prime$):

$F(\tau )\asymp \tau ^p[\log \tau ]^q[\log (\log \tau )]^r$ as $\tau \rightarrow \infty$ for some $p>1$ and q, $r\in { \mathbb {R}}$.

Then, we can show that problem (P) possesses a local-in-time solution if and only if

$$\begin{aligned} \sup _{z\in { \mathbb {R}}^N}\mu (B(z,1))<\infty \end{aligned}$$

in the cases when (i) $p<p_\theta$, (ii) $p=p_\theta$ and $q<-1$, and (iii) $p=p_\theta$, $q=-1$, and $r<-1$. In the other cases, we divide condition (F2$'$) into four cases:

• (1) $p=p_\theta$ and $q=r=-1$.

• (2) $p=p_\theta$, $q=-1$, and $r>-1$.

• (3) $p=p_\theta$, $q>-1$, and $r\in { \mathbb {R}}$.

• (4) $p>p_\theta$ and q, $r\in { \mathbb {R}}$,

and we can identify the optimal singularities of the initial data for solvability of problem (P). Since inclusion of the proofs here would make the paper unduly long, we leave the details to the interested reader.