Supercaloric functions for the parabolic p-Laplace equation in the fast diffusion case

Abstract

We study a generalized class of supersolutions, so-called p-supercaloric functions, to the parabolic p-Laplace equation. This class of functions is defined as lower semicontinuous functions that are finite in a dense set and satisfy the parabolic comparison principle. Their properties are relatively well understood for $p\ge 2$, but little is known in the fast diffusion case $1<p<2$. Every bounded p-supercaloric function belongs to the natural Sobolev space and is a weak supersolution to the parabolic p-Laplace equation for the entire range $1<p<\infty$. Our main result shows that unbounded p-supercaloric functions are divided into two mutually exclusive classes with sharp local integrability estimates for the function and its weak gradient in the supercritical case $\frac{2n}{n+1}<p<2$. The Barenblatt solution and the infinite point source solution show that both alternatives occur. Barenblatt solutions do not exist in the subcritical case $1<p\le \frac{2n}{n+1}$ and the theory is not yet well understood.

1 Introduction

This paper studies classes of supersolutions to the parabolic p-Laplace equation

$$\begin{aligned} \partial _tu -{\text {div}}\left( |\nabla u |^{p-2} \nabla u \right) =0. \end{aligned}$$

(1.1)

The general theory covers the entire parameter range $1<p<\infty$, but different phenomena occur in the slow diffusion case $p>2$ and in the fast diffusion case $1<p<2$. For $p=2$ we have the heat equation. We do not only consider weak solutions, but also weak supersolutions and, more generally, p-supercaloric functions to (1.1). They are pointwise defined lower semicontinuous functions, finite in a dense subset, and are required to satisfy the comparison principle with respect to the solutions of (1.1), see Definition 2.6 below. The definition of supercaloric functions is the same as in classical potential theory for the heat equation when $p=2$, see Watson [23]. By Juutinen et al. [13], the class of p-supercaloric functions is the same as the viscosity supersolutions to (1.1) for $1<p<\infty$. Our results can be extended to more general quasilinear equations

$$\begin{aligned} \partial _tu-{\text {div}}A(x,t,u,\nabla u)=0, \end{aligned}$$

with the p-growth, which are discussed in DiBenedetto [8], DiBenedetto et al. [9] and Wu et al. [24]. For simplicity, we discuss only the prototype case in (1.1).

A p-supercaloric function does not, in general, belong to the natural Sobolev space for (1.1). The only connection to the equation is through the comparison principle. However, Kinnunen and Lindqvist [15] proved that bounded p-supercaloric functions belong to the appropriate Sobolev space and are weak supersolutions to (1.1) for $p \ge 2$. Korte et al. [17] extended the study for a more general class of parabolic equations with p-growth. In this paper we show that bounded p-supercaloric functions are weak solutions to (1.1) for the entire range $1<p<\infty$.

We are mainly interested in unbounded p-supercaloric functions. Assume that u is a p-supercaloric function in $\Omega _T=\Omega \times (0,T)$, where $\Omega$ is an open set in ${\mathbb {R}}^n$ and $T>0$. One of the main results of Kuusi et al. [22] asserts that for $p>2$ there are two mutually exclusive alternatives: Either $u\in L_{{{\,\mathrm{loc}\,}}}^q(\Omega _T)$ for every $0<q<p-1+\frac{p}{n}$ or $u\notin L_{{{\,\mathrm{loc}\,}}}^{p-2}(\Omega _T)$. In particular, if $u\in L_{{{\,\mathrm{loc}\,}}}^{p-2}(\Omega _T)$, then $u\in L_{{{\,\mathrm{loc}\,}}}^q(\Omega _T)$ for every $0<q<p-1+\frac{p}{n}$. For the corresponding theory for the porous medium equation, see Kinnunen et al. [19].

Examples based on the Barenblatt solution (see Barenblatt [3]) and the friendly giant show that both alternatives occur. In the first alternative the upper bound for the exponent is given by the Barenblatt solution

$$\begin{aligned} U(x,t)= (\lambda t)^{-\frac{n}{\lambda }}\left( c-\tfrac{p-2}{p}(\lambda t)^{-\tfrac{p}{\lambda (p-1)}}|x|^{\frac{p}{p-1}}\right) _+^{\frac{p-1}{p-2}}, \quad (x,t)\in {\mathbb {R}}^n\times (0,\infty ),\nonumber \\ \end{aligned}$$

(1.2)

where $2<p<\infty$, $\lambda = n(p-2)+p$ and the constant c is a positive number, which can be chosen such that

$$\begin{aligned} \int _{{\mathbb {R}}^n}U(x,t)\,\, \mathrm {d}x=1 \end{aligned}$$

(1.3)

for every $t>0$. The Barenblatt solution is a weak solution to (1.1) in ${\mathbb {R}}^n\times (0,\infty )$ and the zero extension

$$\begin{aligned} u(x,t)= {\left\{ \begin{array}{ll} U(x,t),&{}\quad t >0,\\ 0, &{}\quad t \le 0, \end{array}\right. } \end{aligned}$$

(1.4)

is p-supercaloric in ${\mathbb {R}}^n \times {\mathbb {R}}$ with $u\in L_{{{\,\mathrm{loc}\,}}}^q({\mathbb {R}}^n \times {\mathbb {R}})$ for every $0<q<p-1+\frac{p}{n}$. This function solves (1.1) with a finite point source, but it fails to belong to the natural Sobolev space, since $|\nabla u|\notin L_{{{\,\mathrm{loc}\,}}}^p({\mathbb {R}}^n \times {\mathbb {R}})$. For the second alternative, we consider a bounded open set $\Omega$ in ${\mathbb {R}}^n$ with a smooth boundary. By separation of variables, we obtain the friendly giant

$$\begin{aligned} U(x,t) =t^{-\frac{1}{p-2}}u(x), \end{aligned}$$

where $u\in C(\Omega )\cap W^{1,p}_{0}(\Omega )$ is a weak solution to the elliptic equation

$$\begin{aligned} {\text {div}}\bigl (|\nabla u|^{p-2}\nabla u\bigr )+\tfrac{1}{p-2}u=0 \end{aligned}$$

in $\Omega$ with $u(x)>0$ for every $x\in \Omega$. The function U is a weak solution to (1.1) in $\Omega \times (0,\infty )$ and the zero extension as in (1.4) is p-supercaloric in $\Omega \times {\mathbb {R}}$ with $u\notin L_{{{\,\mathrm{loc}\,}}}^{p-2}(\Omega \times {\mathbb {R}})$.

We prove the corresponding result in the supercritical range $\frac{2n}{n+1}<p<2$ for a p-supercaloric function u in $\Omega _T$. It asserts that either $u\in L_{{{\,\mathrm{loc}\,}}}^q(\Omega _T)$ for every $0<q<p-1+\frac{p}{n}$ or $u\notin L^{\frac{n}{p}(2-p)}_{{{\,\mathrm{loc}\,}}}(\Omega _T)$. In particular, if $u\in L_{{{\,\mathrm{loc}\,}}}^{\frac{n}{p}(2-p)}(\Omega _T)$, then $u\in L_{{{\,\mathrm{loc}\,}}}^q(\Omega _T)$ for every $0<q<p-1+\frac{p}{n}$. Again, both alternatives occur. For $\frac{2n}{n+1}<p<2$, the Barenblatt solution (see Wu et al. [24, 2.7.2] and Bidaut-Véron [1]) of (1.1) is given by formula

$$\begin{aligned} U(x,t)= (\lambda t)^{-\frac{n}{\lambda }}\left( c+ \tfrac{2-p}{p}(\lambda t)^{-\frac{p}{\lambda (p-1)}}|x|^{\frac{p}{p-1}}\right) ^{-\frac{p-1}{2-p}}, \quad (x,t)\in {\mathbb {R}}^n\times (0,\infty ),\nonumber \\ \end{aligned}$$

(1.5)

where $\lambda = n(p-2)+p$ and the constant c is a positive number so that (1.3) holds for every $t>0$. Observe that $p >\frac{2n}{n+1}$ is equivalent with $\lambda > 0$. Barenblatt solutions do not exist in the subcritical case $1<p\le \frac{2n}{n+1}$ and, to our knowledge, the theory is not yet well understood. The Barenblatt solution is compactly supported for every $t>0$ for $2<p<\infty$, but it is positive everywhere for $\frac{2n}{n+1}<p<2$. As above, the zero extension to the negative times is p-supercaloric in ${\mathbb {R}}^n \times {\mathbb {R}}$ with $u\in L_{{{\,\mathrm{loc}\,}}}^q({\mathbb {R}}^n \times {\mathbb {R}})$ for every $0<q<p-1+\frac{p}{n}$. A prime example of a p-supercaloric function for $\frac{2n}{n+1}<p<2$ that does not belong to the Barenblatt class is the infinite point source solution

$$\begin{aligned} U(x,t)=\left( \frac{ct}{|x|^p}\right) ^{\frac{1}{2-p}}, \quad (x,t)\in {\mathbb {R}}^n\times (0,\infty ), \quad c=(2-p)\left( \tfrac{p}{2-p}\right) ^{p-1}\left( \tfrac{p}{2-p}-n\right) ,\nonumber \\ \end{aligned}$$

(1.6)

see Chasseigne and Vázquez [7]. This function is a solution to (1.1) in $({\mathbb {R}}^n \setminus \{ 0 \})\times (0, \infty )$ and has singularity at $x=0$ for every $t > 0$. Observe that (1.6) is obtained by setting $c=0$ in the Barenblatt solution (1.5) and thus solves (1.1) with an infinite point source. The zero extension as in (1.4) is a p-supercaloric function u in ${\mathbb {R}}^n \times {\mathbb {R}}$ with $u\notin L^{\frac{n}{p}(2-p)}_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n \times {\mathbb {R}})$. Our main result asserts, roughly speaking, that a p-supercaloric function and its gradient have similar local integrability properties than the Barenblatt solution or the corresponding properties are at least as bad as for the infinite point source solution. A Moser type iteration scheme and Harnack estimates are applied in the argument.

2 Weak supersolutions and supercaloric functions

We begin with notation. Let $\Omega \subset {\mathbb {R}}^n$ be an open set. For $T>0$ we denote a space-time cylinders in ${\mathbb {R}}^{n+1}$ by $\Omega _T=\Omega \times (0,T)$ and $\Omega _{t_1, t_2}=\Omega \times (t_1,t_2)$, with $t_1<t_2$. The parabolic boundary of $\Omega _T$ is $\partial _p\Omega _T=({{\overline{\Omega }}}\times \{0\})\cup (\partial \Omega \times (0,T])$. We denote a cube in ${\mathbb {R}}^n$ by $Q=(a_1, b_1)\times \cdots \times (a_n, b_n)$ and the space-time cylinders of the form $Q_{t_1,t_2}$ with $t_1<t_2$, are called boxes. $\Omega '_{t_1,t_2}\Subset \Omega _T$ denotes that $\overline{\Omega '_{t_1,t_2}}$ is a compact subset of $\Omega _T$.

Let $W^{1,p}(\Omega )$ denote the Sobolev space of functions $u\in L^p(\Omega )$, whose first distributional partial derivatives $\frac{\partial u}{\partial x_i}$, $i=1,2,\dots , n$, exist in $\Omega$ and belong to $L^p(\Omega )$. The corresponding Sobolev space with zero boundary values is denoted by $W^{1,p}_0(\Omega )$. The parabolic Sobolev space $L^p(0,T;W^{1,p}(\Omega ))$ consists of functions $u=u(x,t)$ such that for almost every $t \in (0,T)$, the function $x\mapsto u(x,t)$ belongs to $W^{1,p}(\Omega )$ and

$$\begin{aligned} \iint _{\Omega _T} \big ( |u(x,t)|^p+|\nabla u(x,t)|^p \big )\, \, \mathrm {d}x\, \, \mathrm {d}t<\infty . \end{aligned}$$

Observe that the time derivative $\partial _t u$ does not appear anywhere. The definition of the space $L^p(0, T;W_0^{1,p}(\Omega ))$ is similar. We denote $u\in L_{{{\,\mathrm{loc}\,}}}^p(0,T;W^{1,p}_{{{\,\mathrm{loc}\,}}}(\Omega ))$ if $u\in L^p(t_1,t_2;W^{1,p}(\Omega '))$ for every $\Omega '_{t_1,t_2}\Subset \Omega _T$. The definition for $L_{{{\,\mathrm{loc}\,}}}^p({\mathbb {R}};W^{1,p}_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n))$ is analogous. Gradient and divergence are always taken with respect to the spatial variable only.

Weak solutions to (1.1) are assumed to belong to a parabolic Sobolev space, which guarantees a priori local integrability for the function and its weak gradient. We state the definition and results in a space-time cylinder $\Omega _T$, with $T>0$, but extensions to arbitrary cylinders $\Omega _{t_1,t_2}$, with $t_1<t_2$, are obvious.

Definition 2.1

Let $1<p<\infty$ and let $\Omega$ be an open set in ${\mathbb {R}}^n$. A function $u\in L_{{{\,\mathrm{loc}\,}}}^p(0,T;W^{1,p}_{{{\,\mathrm{loc}\,}}}(\Omega ))$ is called a weak solution to (1.1), if

$$\begin{aligned} \iint _{\Omega _T} \left( -u \partial _t \varphi + |\nabla u|^{p-2}\nabla u \cdot \nabla \varphi \right) \,\, \mathrm {d}x\,\, \mathrm {d}t =0 \end{aligned}$$

for every $\varphi \in C^\infty _0(\Omega _T)$. Furthermore, we say that u is a weak supersolution if the integral above is nonnegative for all nonnegative test functions $\varphi \in C^{\infty }_0(\Omega _T)$. If the integral is non-positive for such test functions, we call u a weak subsolution.

Locally bounded weak solutions are locally Hölder continuous, see [8, Chapters III and IV], and locally bounded gradients are locally Hölder continuous, see DiBenedetto [8, Chapter IX]. For lower semicontinuity of weak supersolutions, see Kuusi [21]. The statement of the following existence result can be found in Björn et al. [2, Theorem 2.3]. For the proof, we refer to Ivert [12, Theorem 3.2], which applies Fontes [10]. For related existence results, see Bögelein et al. [5, Theorem 1.2], [6, Theorem 1.2]. Boundary behaviour in more general situations has been studied in Björn et al. [2] and Gianazza et al. [11].

Theorem 2.2

Let $1<p<\infty$ and let $\Omega$ be a bounded open set in ${\mathbb {R}}^n$ with a Lipschitz boundary and $g\in C(\partial _p\Omega _T)$. Then there exists a unique weak solution $u \in C(\overline{\Omega _T})$ to (1.1) with $u=g$ on $\partial _p\Omega _T$. Moreover, if g belongs to $C(0,T;L^2(\Omega ))\cap L^p(0,T;W^{1,p}(\Omega ))$, so does u.

We consider a comparison principle for weak super- and subsolutions. The argument is similar to Kilpeläinen and Lindqvist [14, Lemma 3.1] and Korte et al. [17, Lemma 3.5].

Theorem 2.3

Let $1<p<\infty$ and let $\Omega$ be a bounded open set in ${\mathbb {R}}^n$. Assume that v is a weak supersolution and u is a weak subsolution to (1.1) in $\Omega _T$. If v and $-u$ are lower semicontinuous in ${\overline{\Omega }}_T$ and $u\le v$ on $\partial _p\Omega _T$, then $u\le v$ almost everywhere in $\Omega _T$.

A pointwise limit of a sequence of uniformly bounded weak supersolutions is a weak supersolution, see Korte et al. [17, Theorem 5.3], see also Kinnunen and Lindqvist [15].

Theorem 2.4

Let $1<p<\infty$ and let $\Omega$ be an open set in ${\mathbb {R}}^n$. Assume that $u_i$, $i=1,2,\dots$, are weak supersolutions to (1.1) in $\Omega _T$ such that $\Vert u_i\Vert _{L^\infty (\Omega _T)}\le L<\infty$ for every $i=1,2\dots$ and $u_i\rightarrow u$ almost everywhere in $\Omega _T$ as $i\rightarrow \infty$. Then u is a weak supersolution to (1.1) in $\Omega _T$.

The class of weak supersolutions is closed under taking minimum of two functions, see Korte el al. [17, Lemma 3.2]. See also Kinnunen and Lindqvist [15].

Lemma 2.5

Let $1<p<\infty$ and let $\Omega$ be an open set in ${\mathbb {R}}^n$. If u and v are weak supersolutions to (1.1) in $\Omega _T$, then $\min \{u,v\}$ is a weak supersolution in $\Omega _T$.

The local boundedness assumption in Theorem 2.4 can be replaced with a uniform Sobolev space bound, see Kinnunen and Lindqvist [15] and Korte et al. [17, Remark 5.6]. This implies that if $u\in L_{{{\,\mathrm{loc}\,}}}^p(0,T;W^{1,p}_{{{\,\mathrm{loc}\,}}}(\Omega ))$ and $\min \{u,k\}\in L_{{{\,\mathrm{loc}\,}}}^p(0,T;W^{1,p}_{{{\,\mathrm{loc}\,}}}(\Omega ))$ is a weak supersolution for every $k=1,2,\dots$, then u is a weak supersolution. In general, we have to consider more general class of solutions than weak supersolutions. Thus we define p-supercaloric functions as in Kilpeläinen and Lindqvist [14, Lemma 3.1], see also Kinnunen and Lindqvist [15].

Definition 2.6

Let $1<p<\infty$ and let $\Omega$ be an open set in ${\mathbb {R}}^n$. A function $u :\Omega _T \rightarrow (-\infty ,\infty ]$ is called a p-supercaloric function, if

• (i) u is lower semicontinuous,

• (ii) u is finite in a dense subset, and

• (iii) u satisfies the comparison principle on each box $Q_{t_1,t_2}\Subset \Omega _T$: if $h\in C({\overline{Q}}_{t_1,t_2})$ is a weak solution to (1.1) in $Q_{t_1,t_2}$ and if $h \le u$ on the parabolic boundary of $Q_{t_1,t_2}$, then $h\le u$ in $Q_{t_1,t_2}$.

Similarly, an upper semicontinuous function u is said to be a p-subcaloric function in $\Omega _T$ if $-u$ is p-supercaloric function in $\Omega _T$.

Since p-supercaloric functions are lower semicontinuous, they are locally bounded from below. Thus by adding a constant, we may assume that a p-supercaloric function is nonnegative, when we discuss local properties. If u is a nonnegative p-supercaloric function in $\Omega _T$, the zero extension in the past as in (1.4) is a p-supercaloric function in $\Omega \times (-\infty ,T)$. The following assertion is a direct consequence of the comparison principle in Definition 2.6.

Lemma 2.7

Let $1<p<\infty$ and let $\Omega$ be an open set in ${\mathbb {R}}^n$. If u and v are p-supercaloric functions in $\Omega _T$, then $\min \{u,v\}$ is a p-supercaloric function in $\Omega _T$.

Supercaloric functions are closed under increasing convergence, see [17, Proposition 5.1].

Lemma 2.8

Let $1<p<\infty$ and let $\Omega$ be an open set in ${\mathbb {R}}^n$. Assume that $u_i$, $i=1,2,\dots$, are p-supercaloric functions in $\Omega _T$ such that $u_i\le u_{i+1}$ or every $i=1,2\dots$. If $u=\lim _{i\rightarrow \infty } u$ is finite in a dense set, then u is a p-supercaloric function in $\Omega _T$.

A parabolic comparison principle holds for super- and subcaloric functions, see Kilpeläinen and Lindqvist [14, Lemma 3.1] and Korte et al. [17, Lemma 3.5].

Lemma 2.9

Let $1<p<\infty$ and let $\Omega$ be an open and bounded set in ${\mathbb {R}}^n$. Assume that u is a p-subcaloric function and v is a p-supercaloric function in $\Omega _T$. If $u\le v$ on $\partial _p\Omega _T$, then $u\le v$ in $\Omega _T$.

For the following comparison principle, we refer to Björn et al. [2, Theorem 2.4] and Korte et al. [17, Corollary 4.6]. Observe that the comparison is on the topological boundary instead of the parabolic boundary.

Corollary 2.10

Let $1<p<\infty$ and let U be a bounded open set in ${\mathbb {R}}^{n+1}$. Assume that u is a p-supercaloric function in U and $h\in C({\overline{U}})$ is a weak solution to (1.1) in U. If $h\le u$ on $\{(x,t)\in \partial U:t<T\}$, then $h\le u$ in $\{(x,t)\in U:t<T\}$.

We will apply an existence result for the obstacle problem. In order to guarantee continuity of the solution up to the boundary, we assume that the domain where the obstacle problem is considered has Lipschitz boundary. This condition can be relaxed, see [18, Theorem 3.1].

Theorem 2.11

Let $1<p<\infty$. Assume that $\Omega$ is an open and bounded set in ${\mathbb {R}}^n$ with Lipschitz boundary and let $\psi \in C(\overline{\Omega _T})$. There exists $u \in C(\overline{\Omega _T})$ that is a weak supersolution to (1.1) in $\Omega _T$ with the following properties:

• (i) $u\ge \psi$ in $\Omega _T$ and $u=\psi$ on $\partial _p\Omega _T$,

• (ii) u is a weak solution in the open set $\{(x,t)\in \Omega _T:u(x,t)>\psi (x,t)\}$,

• (iii) u is the smallest weak supersolution above $\psi$, that is, if v is a weak supersolution in $\Omega _T$ and $v\ge \psi$ in $\Omega _T$, then $v\ge u$ in $\Omega _T$.

Proof

For $p>\frac{2n}{n+2}$, see Korte et al. [18, Theorem 3.1]. A careful inspection of the proof reveals that the requirement $p>\frac{2n}{n+2}$ is used to obtain [18, Theorem 2.7] and Theorem 2.2 in the space-time boxes with continuous boundary data which have been used in [18, Construction 3.2]. A proof of [18, Theorem 2.7] for $p>\frac{2n}{n+2}$ is given in Korte et al. [17, Theorem 5.3], see also Kinnunen and Lindqvist [15, Lemma 4.3] for $p>2$. For the corresponding convergence result in the full range $1<p<\infty$ we refer to the Theorem 2.4. $\square$

Next we show that every bounded p-supercaloric function u is a weak supersolution in $\Omega _T$. In this case $u\in L_{{{\,\mathrm{loc}\,}}}^p(0,T;W^{1,p}_{{{\,\mathrm{loc}\,}}}(\Omega ))$ and

$$\begin{aligned} \iint _{\Omega _T} \left( -u \partial _t \varphi + |\nabla u|^{p-2}\nabla u \cdot \nabla \varphi \right) \,\, \mathrm {d}x\,\, \mathrm {d}t\ge 0 \end{aligned}$$

for every nonnegative $\varphi \in C^\infty _0(\Omega _T)$. This result extends Kinnunen and Lindqvist [15, Theorem 1.1] and Korte et al. [17, Theorem 5.8]. The argument is similar to [17, Theorem 5.8], but we repeat it here to show that it applies in the full range $1<p<\infty$.

Theorem 2.12

Let $1<p<\infty$ and let $\Omega$ be an open set in ${\mathbb {R}}^n$. If u is a p-supercaloric function in $\Omega _T$ and u is locally bounded, then u is a weak supersolution to (1.1) in $\Omega _T$.

Proof

Since u is lower semicontinuous, there exists a sequence of continuous functions $\psi _{i}$, $i=1,2,\dots$, such that $\psi _{1}\le \psi _{2}\le \dots \le u$ and

$$\begin{aligned} \lim _{i\rightarrow \infty } \psi _{i}(x,t)= u(x,t) \end{aligned}$$

for every $(x,t)\in \Omega _T$. We claim that u is a weak supersolution in $\Omega _T$. To conclude this it is sufficient to show that u is a weak supersolution in every space-time box $Q_{t_1, t_2}\Subset \Omega _T$.

By Theorem 2.11, for every $i=1,2,\dots$, there exists a solution $v_{i}\in C(\overline{Q_{t_1, t_2}})$ to the obstacle problem in $Q_{t_1, t_2}$ with the obstacle $\psi _{i}$. The function $v_{i}$ is a continuous weak solution in the open set

$$\begin{aligned} U=\{(x,t)\in Q_{t_1, t_2}: v_{i}(x,t)>\psi _{i}(x,t)\}\subset {\mathbb {R}}^{n+1}. \end{aligned}$$

Since $v_{i}=\psi _{i}$ on $\partial _p Q_{t_1, t_2}$, $v_{i}\in C(\overline{Q_{t_1, t_2}})$ and $\psi _{i}\in C(\overline{Q_{t_1, t_2}})$, it follows that $v_{i}=\psi _{i}$ on the boundary $\partial U$, except possibly when $t=t_2$. By Corollary 2.10, we conclude that $u\ge v_{i}$ in U for every $i=1,2,\dots$. Consequently, $\psi _{i}\le v_{i}\le u$ in $Q_{t_1, t_2}$ for every $i=1,2,\dots$ and thus

$$\begin{aligned} \lim _{i\rightarrow \infty }v_{i}(x,t)= u(x,t) \end{aligned}$$

for every $(x,t)\in Q_{t_1, t_2}$. According to the Theorem 2.4, the function u is a weak supersolution in $Q_{t_1, t_2}$. Since this holds true for every $Q_{t_1, t_2}\Subset \Omega _T$ we conclude that u is a weak supersolution in $\Omega _T$. $\square$

3 Barenblatt solutions

This section discusses p-supercaloric functions with a Barenblatt type behaviour in the supercritical range.

Example 3.1

Let $\frac{2n}{n+1}<p<2$. The Barenblatt solution U(x, t) in (1.5) is a weak solution to (1.1) in ${\mathbb {R}}^n\times (0,\infty )$. It follows that the zero extension u(x, t) as in (1.4) is a p-supercaloric function in ${\mathbb {R}}^n\times {\mathbb {R}}$ according to Definition 2.6 and it satisfies

$$\begin{aligned} \iint _{{\mathbb {R}}^n\times {\mathbb {R}}} \left( -u \partial _t \varphi + |\nabla u|^{p-2}\nabla u \cdot \nabla \varphi \right) \,\, \mathrm {d}x\,\, \mathrm {d}t =M\varphi (0), \end{aligned}$$

with $M>0$, for every $\varphi \in C^\infty _0(\Omega _T)$. This means that u solves the equation

$$\begin{aligned} \partial _tu - {\text {div}}(|\nabla u|^{p-2}\nabla u) = M \delta \end{aligned}$$

in the weak sense, where $\delta$ is Dirac’s delta. It follows that the weak gradient $\nabla u(\cdot ,t)$ exists, in the sense of (3.2), for almost every t and it is locally integrable to any power $0<q < p - 1 + \frac{1}{n+1}$. However, the function u is not a weak supersolution to (1.1) in ${\mathbb {R}}^n\times {\mathbb {R}}$, since

$$\begin{aligned} \int _{t_1}^{t_2}\int _{B(0,r)} |\nabla u|^{p-1+\frac{1}{n+1}}\,\, \mathrm {d}x\,\, \mathrm {d}t=\infty \end{aligned}$$

for every $r>0$, $t_1\le 0$ and $t_2>0$. This implies that $u\notin L_{{{\,\mathrm{loc}\,}}}^p({\mathbb {R}};W^{1,p}_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n))$. Observe, that the truncations $\min \{u,k\}$, $k=1,2,\dots$ belong to $L_{{{\,\mathrm{loc}\,}}}^p({\mathbb {R}};W^{1,p}_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n))$ and are weak solutions to (1.1) in ${\mathbb {R}}^{n+1}$ by Theorem 2.12.

Next we recall a Caccioppoli type inequality for nonnegative unbounded weak supersolutions, see Kuusi [20, Lemma 2.2].

Lemma 3.2

Let $1<p<\infty$, $0<\varepsilon <1$ and let $\Omega$ be an open set in ${\mathbb {R}}^n$. Assume that u is a nonnegative weak supersolution in $\Omega _T$. There exists a constant $c = c(p,\varepsilon )$ such that

$$\begin{aligned} \iint _{\Omega _T}&|\nabla u|^pu^{-\varepsilon -1}\varphi ^p\,\, \mathrm {d}x\, \, \mathrm {d}t + {{\,\mathrm{ess\,sup}\,}}_{0<t<T} \int _\Omega u^{1-\varepsilon } \varphi ^p\, \, \mathrm {d}x \\&\le c \iint _{\Omega _T}u^{p-1-\varepsilon } |\nabla \varphi |^p \, \, \mathrm {d}x\, \, \mathrm {d}t + c\iint _{\Omega _T}u^{1-\varepsilon } |\partial _t (\varphi ^p)| \,\, \mathrm {d}x\, \, \mathrm {d}t \end{aligned}$$

for every nonnegative test function $\varphi \in C_0^\infty (\Omega _T)$.

The following version of Sobolev’s inequality will be useful for us, see DiBenedetto [8, Proposition 3.1, p. 7] and DiBenedetto et al. [9, Proposition 4.1].

Lemma 3.3

Let $0<m<\infty$ and $1\le p<\infty$. Assume that $u\in L^p_{{{\,\mathrm{loc}\,}}}(0,T;W_{{{\,\mathrm{loc}\,}}}^{1,p}(\Omega ))$ and $\varphi \in C^\infty _0(\Omega _T)$. Then there exists a constant $c=c(p,m,n)$ such that

$$\begin{aligned} \iint _{\Omega _T}|\varphi u|^q \, \mathrm {d}x\, \mathrm {d}t\le c\iint _{\Omega _T}|\nabla (\varphi u)|^p\, \mathrm {d}x \, \mathrm {d}t\left( {{\,\mathrm{ess\,sup}\,}}_{0<t<T}\int _\Omega |\varphi u|^m \, \mathrm {d}x\right) ^{\frac{p}{n}}, \end{aligned}$$

where $q=p+\frac{pm}{n}$.

We prove a general local integrability result for unbounded p-supercaloric functions. The obtained integrability exponent is sharp as shown by the Barenblatt solution. We apply a Moser type iteration scheme.

Theorem 3.4

Let $\frac{2n}{n+1}<p<2$ and let $\Omega$ be an open set in ${\mathbb {R}}^n$. Assume that u is a p-supercaloric function in $\Omega _T$. If $u \in L_{{{\,\mathrm{loc}\,}}}^{s}(\Omega _T)$ for some $s > \frac{n}{p}(2-p)$, then $u\in L_{{{\,\mathrm{loc}\,}}}^q(\Omega _T)$ whenever $0<q<p-1+\frac{p}{n}$.

Proof

Since u is locally bounded from below, by adding a constant, we may assume that $u\ge 1$. We may assume $0<s < 1$, otherwise we have $u\in L_{{{\,\mathrm{loc}\,}}}^1(\Omega _T)$ and we can directly proceed to the last step of the proof.

Since $p>\frac{2n}{n+1}$, we have $\frac{n}{p}(2-p)<1$. We consider the truncations $u_k=\min \{u, k\}$, $k=2, 3,\dots$. By Theorem 2.12, the function $u_k$ is a weak supersolution to (1.1) and it satisfies the Caccioppoli estimate in Lemma 3.2. Let $\varphi \in C_0^\infty (\Omega _T)$, $0\le \varphi \le 1$ and $\varphi =1$ in a compact subset of $\Omega _T$. For the first step of iteration, we choose $0<\varepsilon <1$ in Lemma 3.2 so that $1-\varepsilon =s$. Since $u_k\ge 1$, we obtain

$$\begin{aligned} \iint _{\Omega _T}\left| \nabla \left( \varphi u_k^{\frac{s-(2-p)}{p}}\right) \right| ^p\,\, \mathrm {d}x\,\, \mathrm {d}t&\le c(p,s)\left( \iint _{\Omega _T}u_k^{s-(2-p)}|\nabla \varphi |^p\,\, \mathrm {d}x\,\, \mathrm {d}t \right. \\&\quad \left. + \iint _{\Omega _T} u_k^{s-2}|\nabla u_k|^p\varphi ^p\,\, \mathrm {d}x\,\, \mathrm {d}t\right) \\&\le c(p,s)\left( \iint _{\Omega _T}u_k^{s-(2-p)}|\nabla \varphi |^p\,\, \mathrm {d}x\,\, \mathrm {d}t \right. \\&\quad \left. + \iint _{\Omega _T} u_k^{s}\left| \partial _t(\varphi ^p)\right| \,\, \mathrm {d}x\,\, \mathrm {d}t\right) \\&\le c(p,s)\iint _{\Omega _T}u_k^{s}\left( |\nabla \varphi |^p+ \left| \partial _t(\varphi ^p)\right| \right) \, \, \mathrm {d}x\,\, \mathrm {d}t\\&\le c(p,s)\iint _{\Omega _T}u^{s}\left( |\nabla \varphi |^p+\left| \partial _t(\varphi ^p)\right| \right) \,\, \mathrm {d}x\,\, \mathrm {d}t<\infty , \end{aligned}$$

for every $k=2,3,\dots$. By Lemma 3.3, we have

$$\begin{aligned} \iint _{\Omega _T} \varphi ^q u_k^{s-(2-p)+\frac{s p}{n}}\,\, \mathrm {d}x\,\, \mathrm {d}t&=\iint _{\Omega _T}\left| \varphi u_k^{\frac{s-(2-p)}{p}}\right| ^q \,\, \mathrm {d}x\,\, \mathrm {d}t \\&\le c(p,s,n)\iint _{\Omega _T}\left| \nabla \left( \varphi u_k^{\frac{s-(2-p)}{p}} \right) \right| ^p\,\\&\quad \, \mathrm {d}x\,\, \mathrm {d}t \left( {{\,\mathrm{ess\,sup}\,}}_{0<t<T}\int _\Omega \varphi ^m u_k^s\,\, \mathrm {d}x\right) ^{\frac{p}{n}}, \end{aligned}$$

with $m=\frac{ps}{s-(2-p)}$ and $q=p+\frac{pm}{n}$. Observe that

$$\begin{aligned} \varphi (x,t)^m=\left( \varphi (x,t)^{\frac{s}{s-(2-p)}} \right) ^p, \end{aligned}$$

with $\frac{m}{p} > 1$ so that $\varphi ^{\frac{m}{p}} \in C^1_0(\Omega _T)$, and thus we may apply Lemma 3.2 for second term as well. We have

$$\begin{aligned} {{\,\mathrm{ess\,sup}\,}}_{0<t<T}\int _\Omega \varphi ^m u_k^s\,\, \mathrm {d}x&\le c(p,s)\\ {}&\left( \iint _{\Omega _T}u_k^{s-(2-p)}|\nabla (\varphi ^{\frac{m}{p}})|^p\,\, \mathrm {d}x\,\, \mathrm {d}t + \iint _{\Omega _T} u_k^{s}\left| \partial _t(\varphi ^m)\right| \,\, \mathrm {d}x\,\, \mathrm {d}t\right) \\&\le c(p,s)\iint _{\Omega _T}u^{s}\left( |\nabla (\varphi ^{\frac{m}{p}})|^p+\left| \partial _t(\varphi ^{m})\right| \right) \,\, \mathrm {d}x\,\, \mathrm {d}t\\&\le c(p,s)\iint _{\Omega _T}u^{s}\varphi ^{m-p}\left( |\nabla \varphi |^p+\left| \partial _t(\varphi ^p)\right| \right) \,\, \mathrm {d}x\,\, \mathrm {d}t<\infty \end{aligned}$$

for every $k=2,3,\dots$. By combining the estimates and applying Lebesgue’s monotone convergence theorem we conclude

$$\begin{aligned} \iint _{\Omega _T} \varphi ^q u^{s-(2-p)+\frac{s p}{n}}\,\, \mathrm {d}x\,\, \mathrm {d}t =\lim _{k\rightarrow \infty }\iint _{\Omega _T} \varphi ^q u_k^{s-(2-p)+\frac{s p}{n}}\,\, \mathrm {d}x\,\, \mathrm {d}t <\infty . \end{aligned}$$

Since $s >\frac{n}{p}(2-p)$, it follows that $s-(2-p)+\frac{s p}{n}> s$. Thus $u\in L_{{{\,\mathrm{loc}\,}}}^{s(1+\frac{p}{n})-(2-p)}(\Omega _T)$.

Denote $s_0= s$ and $s_1=s_0(1+\frac{p}{n})-(2-p)$. After the first step of iteration we have $u\in L_{{{\,\mathrm{loc}\,}}}^{s_1}(\Omega _T)$. If $s_1 <1$, then in the second step of iteration we choose $1-\varepsilon = s_1$ and again combine the Caccioppoli inequality and the Sobolev inequality with $m = \frac{p s_1}{s_1-(2-p)}$. We continue in this way and obtain an increasing sequence of numbers $s_i$, satisfying

$$\begin{aligned} s_i = s_{i-1}\left( 1+\tfrac{p}{n}\right) - (2-p), \qquad i=1,2,\dots , \end{aligned}$$

which can be written in terms of $s_0$ as

$$\begin{aligned} s_i =\left( 1+\tfrac{p}{n}\right) ^i\left( s_0-\tfrac{n}{p}(2-p)\right) +\tfrac{n}{p}(2-p), \qquad i=0,1,2,\dots . \end{aligned}$$

After the ith step of iteration we have $u\in L_{{{\,\mathrm{loc}\,}}}^{s_i}(\Omega _T)$. After a finite number of iterations we have $s_i\ge 1$ and thus $u\in L_{{{\,\mathrm{loc}\,}}}^1(\Omega _T)$.

In order to pass from $u\in L_{{{\,\mathrm{loc}\,}}}^1(\Omega _T)$ to $u\in L_{{{\,\mathrm{loc}\,}}}^{p-1+\frac{p}{n}-\sigma } (\Omega _T)$ for every $\sigma >0$ we apply a similar argument once more. Let $\varepsilon = \frac{\sigma }{1+\frac{p}{n}}$. Then $p-1-\varepsilon +\tfrac{p(1-\varepsilon )}{n} =p-1+\tfrac{p}{n}-\sigma$. By Lemma 3.3, we have

$$\begin{aligned} \iint _{\Omega _T} \varphi ^q u_k^{p-1+\frac{p}{n}-\sigma }\,\, \mathrm {d}x\,\, \mathrm {d}t&=\iint _{\Omega _T} \varphi ^q u_k^{p-1-\varepsilon +\frac{p(1-\varepsilon )}{n}}\,\, \mathrm {d}x\,\, \mathrm {d}t =\iint _{\Omega _T}\left| \varphi u_k^{\frac{p-1-\varepsilon }{p}}\right| ^q \,\, \mathrm {d}x\,\, \mathrm {d}t \\&\le c(p,\varepsilon ,n)\iint _{\Omega _T}\left| \nabla \left( \varphi u_k^{\frac{p-1-\varepsilon }{p}} \right) \right| ^p\,\\&\quad \, \mathrm {d}x\,\, \mathrm {d}t \left( {{\,\mathrm{ess\,sup}\,}}_{0<t<T}\int _\Omega \varphi ^m u_k^{1-\varepsilon }\,\, \mathrm {d}x\right) ^{\frac{p}{n}}, \end{aligned}$$

with $m=\frac{p(1-\varepsilon )}{p-1-\varepsilon }$ and $q=p+\frac{pm}{n}$.

By Lemma 3.2, with $u_k\ge 1$, this implies

$$\begin{aligned} \begin{aligned} \iint _{\Omega _T}&\left| \nabla \left( \varphi u_k^{\frac{p-1-\varepsilon }{p}}\right) \right| ^p\,\, \mathrm {d}x\,\, \mathrm {d}t\\&\le c(p,\varepsilon )\left( \iint _{\Omega _T}u_k^{p-1-\varepsilon }|\nabla \varphi |^p\,\, \mathrm {d}x\,\, \mathrm {d}t + \iint _{\Omega _T} u_k^{-\varepsilon -1}|\nabla u_k|^p\varphi ^p\,\, \mathrm {d}x\,\, \mathrm {d}t\right) \\&\le c(p,\varepsilon )\left( \iint _{\Omega _T}u_k^{p-1-\varepsilon }|\nabla \varphi |^p\,\, \mathrm {d}x\,\, \mathrm {d}t + \iint _{\Omega _T} u_k^{p-1-\varepsilon }\left| \partial _t(\varphi ^p)\right| \,\, \mathrm {d}x\,\, \mathrm {d}t\right) \\&\le c(p,\varepsilon )\iint _{\Omega _T}u_k^{1-\varepsilon }\left( |\nabla \varphi |^p+ \left| \partial _t(\varphi ^p)\right| \right) \, \, \mathrm {d}x\,\, \mathrm {d}t\\&\le c(p,\varepsilon )\iint _{\Omega _T}u\left( |\nabla \varphi |^p+\left| \partial _t(\varphi ^p)\right| \right) \,\, \mathrm {d}x\,\, \mathrm {d}t<\infty \end{aligned} \end{aligned}$$

(3.1)

for every $k=2,3,\dots$. For the second term, Lemma 3.2 implies

$$\begin{aligned} {{\,\mathrm{ess\,sup}\,}}_{0<t<T}\int _\Omega \varphi ^m u_k^{1-\varepsilon }\,\, \mathrm {d}x&\le c(p,\varepsilon )\left( \iint _{\Omega _T}u_k^{p-1-\varepsilon }|\nabla (\varphi ^{\frac{m}{p}})|^p\,\, \mathrm {d}x\,\, \mathrm {d}t \right. \\&\quad \left. + \iint _{\Omega _T} u_k^{p-1-\varepsilon }\left| \partial _t(\varphi ^m)\right| \,\, \mathrm {d}x\,\, \mathrm {d}t\right) \\&\le c(p,\varepsilon )\iint _{\Omega _T}u^{1-\varepsilon }\left( |\nabla (\varphi ^{\frac{m}{p}})|^p+\left| \partial _t(\varphi ^{m})\right| \right) \,\, \mathrm {d}x\,\, \mathrm {d}t\\&\le c(p,\varepsilon )\iint _{\Omega _T}u^{1-\varepsilon }\varphi ^{m-p}\left( |\nabla \varphi |^p+\left| \partial _t(\varphi ^p)\right| \right) \,\, \mathrm {d}x\,\, \mathrm {d}t\\&\le c(p,\varepsilon )\iint _{\Omega _T}u\left( |\nabla \varphi |^p+\left| \partial _t(\varphi ^p)\right| \right) \,\, \mathrm {d}x\,\, \mathrm {d}t <\infty \end{aligned}$$

for every $k=2,3,\dots$. Thus

$$\begin{aligned} \iint _{\Omega _T} \varphi ^q u^{p-1+\frac{p}{n}-\sigma }\,\, \mathrm {d}x\,\, \mathrm {d}t =\lim _{k\rightarrow \infty }\iint _{\Omega _T} \varphi ^q u_k^{p-1+\frac{p}{n}-\sigma }\,\, \mathrm {d}x\,\, \mathrm {d}t<\infty , \end{aligned}$$

which implies $u\in L_{{{\,\mathrm{loc}\,}}}^{p-1+\frac{p}{n}-\sigma } (\Omega _T)$. $\square$

Let u be a nonnegative p-supercaloric function in $\Omega _T$. It may happen that u does not have a locally integrable weak derivative. In this case we consider the truncations $u_k=\min \{u, k\}\in L^p_{{{\,\mathrm{loc}\,}}}(\Omega _T)$, $k=1,2,\dots$ and define the weak gradient by

$$\begin{aligned} \nabla u = \lim _{k\rightarrow \infty } \nabla u_k, \end{aligned}$$

(3.2)

which is a well defined measurable function but does not necessarily belong to $L^1_{{{\,\mathrm{loc}\,}}}(\Omega _T)$. If $\nabla u\in L^1_{{{\,\mathrm{loc}\,}}}(\Omega _T)$, then $\nabla u(\cdot ,t)$ is the Sobolev gradient of $u(\cdot ,t)$ for almost every t with $0<t<T$.

Theorem 3.5

Let $\frac{2n}{n+1}<p<2$ and let $\Omega$ be an open set in ${\mathbb {R}}^n$. Assume that u is a p-supercaloric function in $\Omega _T$ with $u \in L_{{{\,\mathrm{loc}\,}}}^{s}(\Omega _T)$ for some $s > \frac{n}{p}(2-p)$, Then $\nabla u\in L^q_{{{\,\mathrm{loc}\,}}}(\Omega _T)$ whenever $0<q<p-1+\frac{1}{n+1}$.

Proof

Since u is locally bounded from below, by adding a constant, we may assume that $u\ge 1$. Let $0<t_1<t_2<T$, $\Omega ' \Subset \Omega$ and $\varepsilon \in (0,1)$. We consider the truncations $u_k=\min \{u, k\}$, $k=1,2,\dots$. By Hölder’s inequality, we have

$$\begin{aligned} \int _{t_1}^{t_2} \int _{\Omega ' }&|\nabla u_k|^q \,\, \mathrm {d}x\,\, \mathrm {d}t = \int _{t_1}^{t_2} \int _{\Omega ' } \left( u_k^{-\frac{1+\varepsilon }{p}} |\nabla u_k| \right) ^q u_k^{q\frac{1+\varepsilon }{p}} \,\, \mathrm {d}x\,\, \mathrm {d}t \\&\le \left( \int _{t_1}^{t_2} \int _{\Omega ' } u_k^{-1-\varepsilon } |\nabla u_k|^p\, \, \mathrm {d}x \,\, \mathrm {d}t \right) ^\frac{q}{p} \left( \int _{t_1}^{t_2} \int _{\Omega ' } u_k^{q\frac{1+\varepsilon }{p-q}}\,\, \mathrm {d}x\,\, \mathrm {d}t\right) ^{1-\frac{q}{p}} \\&= \left( \frac{p}{p-1-\varepsilon } \right) ^q \left( \int _{t_1}^{t_2} \int _{\Omega ' } \left| \nabla \left( u_k^\frac{p-1-\varepsilon }{p} \right) \right| ^p \, \, \mathrm {d}x\, \, \mathrm {d}t \right) ^\frac{q}{p} \left( \int _{t_1}^{t_2} \int _{\Omega ' } u_k^{q\frac{1+\varepsilon }{p-q}} \, \, \mathrm {d}x\, \, \mathrm {d}t \right) ^{1-\frac{q}{p}}, \end{aligned}$$

where the first integral is uniformly bounded with respect to k as in (3.1) and the second integral is finite whenever $q\frac{1+\varepsilon }{p-q} < p - 1 + \frac{p}{n}$ by Theorem 3.4. From this we can conclude that the right hand side is finite for any $0<q < p - 1 + \frac{1}{n+1}$. Now we have that $\nabla u_k$ is uniformly bounded in $L_{{{\,\mathrm{loc}\,}}}^q(\Omega _T)$. $\square$

Remark 3.6

If $p>\frac{2n+1}{n+1}$, then $p - 1 + \frac{1}{n+1}>1$ and it follows that $\nabla u(\cdot ,t)$ in Lemma 3.5 is the Sobolev gradient for almost every t with $0<t<T$. However, if $\frac{2n}{n+1} < p \le \frac{2n+1}{n+1}$, then $p-1+\frac{1}{n+1} \le 1$. In this case the Sobolev gradient may not exist and the weak gradient $\nabla u$ is interpreted as in (3.2).

Remark 3.7

Let $\frac{2n}{n+1}<p<2$ and let $\Omega$ be an open set in ${\mathbb {R}}^n$. Assume that u is a p-supercaloric function in $\Omega _T$ with $u \in L_{{{\,\mathrm{loc}\,}}}^{s}(\Omega _T)$ for some $s > \frac{n}{p}(2-p)$. By Theorem 3.5 we have $\nabla u\in L^{p-1}_{{{\,\mathrm{loc}\,}}}(\Omega _T)$. Theorem 2.12 implies

$$\begin{aligned} \iint _{\Omega _T}&\left( - u \partial _t \varphi + |\nabla u|^{p-2}\nabla u \cdot \nabla \varphi \right) \, \, \mathrm {d}x\, \, \mathrm {d}t \\&=\lim _{k\rightarrow \infty }\iint _{\Omega _T} \left( - u_k \partial _t \varphi + |\nabla u_k|^{p-2}\nabla u_k \cdot \nabla \varphi \right) \, \, \mathrm {d}x\, \, \mathrm {d}t \ge 0 \end{aligned}$$

for every nonnegative $\varphi \in C_0^\infty (\Omega _T)$. By the Riesz representation theorem there exists a nonnegative Radon measure $\mu$ on ${\mathbb {R}}^{n+1}$ such that

$$\begin{aligned} \iint _{\Omega _T}\left( - u \partial _t \varphi + |\nabla u|^{p-2}\nabla u \cdot \nabla \varphi \right) \, \, \mathrm {d}x\, \, \mathrm {d}t =\iint _{\Omega _T}\varphi \,\, \mathrm {d}\mu \end{aligned}$$

for every $\varphi \in C_0^\infty (\Omega _T)$. This means that u is a solution to the measure data problem

$$\begin{aligned} \partial _tu -{\text {div}}\left( |\nabla u |^{p-2} \nabla u \right) =\mu . \end{aligned}$$

Note that $\nabla u\notin L^{p}_{{{\,\mathrm{loc}\,}}}(\Omega _T)$, in general. The integrability of the gradient of a weak solution to a measure data problem in the fast diffusion case has been studied Baroni [4].

Remark 3.8

Let $\frac{2n}{n+1}<p<2$ and let $\Omega$ be an open set in ${\mathbb {R}}^n$. Assume that u is a nonnegative p-supercaloric function in $\Omega _T$ with $u \in L_{{{\,\mathrm{loc}\,}}}^{2-p}(\Omega _T)$. Then $\nabla \log u \in L^p_{{{\,\mathrm{loc}\,}}}(\Omega _T)$, where $\nabla$ stands for the usual Sobolev gradient. This property holds for u satisfying the assumptions in Theorem 3.4, since $2-p < \frac{n}{p}(2-p)$. The logarithmic estimate can be obtained as in [16] by taking $\varepsilon = p-1$ in Lemma 3.2.

Remark 3.9

By the following dichotomy it follows that p-supercaloric function is either locally integrable to any power smaller than $p -1 + \frac{p}{n} > 1$, or it is not integrable to a power $\frac{n}{p}(2-p) < 1$. Observe that as $p \searrow \frac{2n}{n+1}$ we have $\frac{n}{p}(2-p) \nearrow 1$ and $p -1 + \frac{p}{n} \searrow 1$. The gap between the exponents becomes smaller and shrinks to a point as p approaches the critical value from above.

We have the following characterization for Barenblatt type p-supercaloric functions.

Theorem 3.10

Let $\frac{2n}{n+1}<p<2$ and let $\Omega$ be an open set in ${\mathbb {R}}^n$. Assume that u is a p-supercaloric function in $\Omega _T$. Then the following assertions are equivalent:

• (i) $u\in L^q_{{{\,\mathrm{loc}\,}}}(\Omega _T)$ for some $q> \frac{n}{p}(2-p)$,

• (ii) $u \in L^{\frac{n}{p}(2-p)}_{{{\,\mathrm{loc}\,}}} (\Omega _T)$,

• (iii’) there exists $\alpha \in (\frac{n}{p}(2-p), 1)$ such that

  $$\begin{aligned} {{\,\mathrm{ess\,sup}\,}}_{\delta<t<T-\delta } \int _{\Omega '} |u(x,t)|^\alpha \,\, \mathrm {d}x<\infty , \end{aligned}$$

  whenever $\Omega '\Subset \Omega$ and $\delta \in (0, \frac{T}{2})$,

• (iii)
  $$\begin{aligned} {{\,\mathrm{ess\,sup}\,}}_{\delta<t<T-\delta } \int _{\Omega '} |u(x,t)|\,\, \mathrm {d}x<\infty , \end{aligned}$$

  whenever $\Omega '\Subset \Omega$ and $\delta \in (0, \frac{T}{2})$.

Proof

Since u is locally bounded from below, by adding a constant, we may assume that $u\ge 1$.

(i) $\implies$ (ii): Direct consequence of Hölder’s inequality.

(iii) $\implies$ (iii’): Hölder’s inequality. (iii’) $\implies$ (i): Follows from the iteration argument in the proof of Theorem 3.4.

(i) $\implies$ (iii’): Let $\Omega '\Subset \Omega$ and $\delta \in (0, \frac{T}{2})$. We consider the truncations $u_k=\min \{u, k\}$, $k=2,3,\dots$. Since $u_k$ is a weak supersolution in $\Omega _T$, it satisfies the Caccioppoli inequality in Lemma 3.2. Choose $\varepsilon >0$ small enough such that $1-\varepsilon = q$. By the assumption, $u\in L^{1-\varepsilon }_{{{\,\mathrm{loc}\,}}}(\Omega _T)$. By choosing a test function $\varphi \in C_0^\infty (\Omega _T)$ with $0\le \varphi \le 1$ and $\varphi =1$ in $\Omega '\times (\delta , T-\delta )$ in Lemma 3.2, we have

$$\begin{aligned} {{\,\mathrm{ess\,sup}\,}}_{\delta<t<T-\delta }\int _{\Omega '} u_k^{1-\varepsilon } \,\, \mathrm {d}x&\le c(p,\varepsilon )\left( \iint _{\Omega _T} u^{p-1-\varepsilon }_k|\nabla \varphi |^p\,\, \mathrm {d}x\,\, \mathrm {d}t +\iint _{\Omega _T} u^{1-\varepsilon }_k\left| \partial _t (\varphi ^p)\right| \,\, \mathrm {d}x\,\, \mathrm {d}t\right) \\&\le c(p,\varepsilon )\iint _{\Omega _T} u^{1-\varepsilon }_k\left( |\nabla \varphi |^p+ \left| \partial _t (\varphi ^p)\right| \right) \, \mathrm {d}x\,\, \mathrm {d}t\\&\le c(p,\varepsilon )\iint _{\Omega _T} u^{1-\varepsilon }\left( |\nabla \varphi |^p+ \left| \partial _t(\varphi ^p)\right| \right) \, \mathrm {d}x\,\, \mathrm {d}t. \end{aligned}$$

The claim follows by letting $k\rightarrow \infty$.

(ii) $\implies$ (i): Follows by showing contraposition $\lnot$(i) $\implies$ $\lnot$(ii) by using Theorem 4.6.

(i) $\implies$ (iii): Follows by showing contraposition $\lnot$(iii) $\implies$ $\lnot$(i) by using Theorem 4.6. $\square$

Remark 3.11

$\nabla u\in L^q_{{{\,\mathrm{loc}\,}}}(\Omega _T)$, whenever $0<q<p-1+\frac{1}{n+1}$, is another equivalent assertion in Theorem 3.10 for $p >\frac{2n+1}{n+1}$. This follows from the fact that $\nabla u$ is a weak gradient in Sobolev’s sense and $u \in L^1_{{{\,\mathrm{loc}\,}}} (\Omega _T)$. By the proof of Theorem 3.4, this implies (i) in Theorem 3.10.

4 Infinite point source solutions

This section discusses p-supercaloric functions in the supercritical case that are not covered by Theorem 3.10. The leading example is the infinite point source solution in (1.6). In contrast with the friendly giant in the slow diffusion case $p>2$, it is the singularity in space, not in time, that fails to be locally integrable to an appropriate power. Roughly speaking, at every time slice the singularity of the infinite point source solution is worse power type singularity than

$$\begin{aligned} u(x,t)=|x|^{- \frac{n-p}{p-1}}, \quad 1<p<2, \end{aligned}$$

based on the fundamental solution to the elliptic p-Laplace equation.

Example 4.1

Let $\frac{2n}{n+1}<p<2$. The infinite point source solution U(x, t) in (1.6) is a weak solution to (1.1) in ${\mathbb {R}}^n\times (0,\infty )$. It follows that the zero extension u(x, t) as in (1.4) is a p-supercaloric function in ${\mathbb {R}}^n\times {\mathbb {R}}$. However, u(x, t) is not a weak supersolution to (1.1) in any domain which contains a neighbourhood of $x = 0$, since

$$\begin{aligned} \int _{t_1}^{t_2}\int _{B(0,r)} |\nabla u|^{\frac{n}{2}(2-p)}\,\, \mathrm {d}x\,\, \mathrm {d}t=\infty , \end{aligned}$$

for every $r>0$, $t_1>-\infty$ and $t_2>0$. This implies that $u\notin L_{{{\,\mathrm{loc}\,}}}^p({\mathbb {R}};W^{1,p}_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n))$. Observe, that the truncations $\min \{u,k\}$, $k=1,2,\dots$ belong to $L_{{{\,\mathrm{loc}\,}}}^p({\mathbb {R}};W^{1,p}_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n))$ and are weak solutions to (1.1) by Theorem 2.12.

Example 4.2

Let $\frac{2n}{n+1}<p<2$. First show that there exist supercaloric functions that are not locally integrable to any positive power. Consider

$$\begin{aligned} u(x,t)= \left( \frac{ct}{|x|^q} \right) ^\frac{1}{2-p}, \end{aligned}$$

where $q \ge p$ and the constant c will be determined later. We show that u is a supersolution to (1.1) in $(B(0,1) \setminus \{0\}) \times (0,\infty )$. A direct computation shows that

$$\begin{aligned} \partial _t u (x,t)= \tfrac{c^\frac{1}{2-p}}{2-p} t^\frac{p-1}{2-p} |x|^{-\frac{q}{2-p}} \quad \text {and}\quad \nabla u(x,t) = - \tfrac{q}{2-p} (ct)^\frac{1}{2-p} |x|^{-\frac{q}{2-p}-1} \frac{x}{|x|}. \end{aligned}$$

It follows that

$$\begin{aligned} {\text {div}}\left( |\nabla u(x,t) |^{p-2} \nabla u(x,t) \right) = \left( \tfrac{q}{2-p} \right) ^{p-1} \left( \tfrac{q}{2-p} -q + p -n\right) (ct)^\frac{p-1}{2-p} |x|^{q-p-\frac{q}{2-p}}. \end{aligned}$$

It is easy to see that $\frac{q}{2-p} -q + p -n > 0$ for any $q \ge p$, and $\frac{q}{2-p} \ge -q + p + \frac{q}{2-p}$, so that $|x|^{q-p-\frac{q}{2-p}} \le |x|^{-\frac{q}{2-p}}$ for every $x\in B(0,1)$. By choosing

$$\begin{aligned} c=c(n,p,q)= (2-p) \left( \tfrac{q}{2-p} \right) ^{p-1} \left( \tfrac{q}{2-p} - q + p -n \right) , \end{aligned}$$

(4.1)

we have

$$\begin{aligned} \partial _t u(x,t) \ge {\text {div}}\left( |\nabla u(x,t) |^{p-2} \nabla u(x,t) \right) \end{aligned}$$

for every $(x,t)\in B(0,1) \times (0,\infty )$. Lemma 2.8 implies that u is a p-supercaloric function in $B(0,1) \times (0,\infty )$. For any $\varepsilon > 0$, by choosing $q > \frac{n}{\varepsilon }(2-p)$, we obtain a p-supercaloric function $u \notin L^{\varepsilon }_{{{\,\mathrm{loc}\,}}}(B(0,1) \times (0,\infty ))$. This shows that, in general, a p-supercaloric function is not locally integrable to any positive power. The same example with the constant in (4.1) is a supersolution to (1.1) also for $1<p\le \frac{2n}{n+1}$, with the additional requirement $q > \frac{(n-p)(2-p)}{p-1}$. The function u is p-supercaloric for any $c > 0$ with $0<q \le \frac{(n-p)(2-p)}{p-1}$ and $1<p<2$, since then

$$\begin{aligned} \partial _t u(x,t) \ge 0 \ge {\text {div}}\left( |\nabla u(x,t) |^{p-2} \nabla u(x,t) \right) \end{aligned}$$

for $(x,t) \in {\mathbb {R}}^n\times (0,\infty )$.

The following intrinsic weak Harnack inequality for supersolutions can be found in [11, Proposition 3.1].

Lemma 4.3

Let $\frac{2n}{n+1}<p<2$ and let $\Omega$ be an open set in ${\mathbb {R}}^n$. Assume that u is a nonnegative lower semicontinuous weak supersolution to (1.1) in $\Omega _T$. There exist constants $c_1=c_1(n,p) \in (0,1)$ and $c_2=c_2(n,p) \in (0,1)$ such that, for almost every $s \in (0,T)$, we have

for every $t \in [s+\frac{3}{4}\theta r^p, s+\theta r^p]$, with

whenever $B(x_0,16r) \times [s,s+\theta r^p] \subset \Omega _T$.

We shall also apply the following Harnack inequality for weak solutions, see [9, Appendix A, Proposition A.1.1].

Lemma 4.4

Let $1<p<2$ and let $\Omega$ be an open set in ${\mathbb {R}}^n$. Assume that h is a nonnegative continuous weak solution to (1.1) in $\Omega _T$. There exists a constant $c = c(p,n)$ such that

whenever $B(x_0,2 r) \times [s,t] \subset \Omega _T$.

Next lemma will be a useful building block in the characterization of the complementary class of the Barenblatt type supercaloric functions.

Lemma 4.5

Let $\frac{2n}{n+1}<p<2$ and let $\Omega$ be an open set in ${\mathbb {R}}^n$. Assume that u is a nonnegative supercaloric function in $\Omega _T$ and let $(x_0,t_0) \in \Omega _T$. Let $0<t_j<T$, $j=1,2,\dots$, with $t_j \rightarrow t_0$ as $j \rightarrow \infty$. If for every $r_0 > 0$ there exists $0<r \le r_0$ such that

$$\begin{aligned} \lim _{j\rightarrow \infty } \int _{B(x_0,r)} u(x,t_j) \, \, \mathrm {d}x = \infty , \end{aligned}$$

(4.2)

then

$$\begin{aligned} \liminf _{\begin{array}{c} (x,t)\rightarrow (x_0,s) \\ t>s \end{array} } u(x,t)|x-x_0|^{\frac{p}{2-p}}>0 \end{aligned}$$

for every $s > t_0$.

Proof

First we observe that (4.2) implies that

$$\begin{aligned} \lim _{j\rightarrow \infty } \int _{B(x_0,r)} u(x,t_j) \, \, \mathrm {d}x = \infty \end{aligned}$$

for every $r > 0$. We construct a Poisson modification and apply Lemma 4.4 for the obtained solution and Lemma 4.3 for the truncations $u_k=\min \{u,k\}$, $k=1,2,\dots$, which are weak supersolutions by Theorem 2.12.

Let $t \in (t_0,T)$ and $r > 0$. The assumption in (4.2) implies that

for j large enough. This implies that, when j is large enough, we may choose a real number $k_j$ such that

(4.3)

We would like to obtain a weak solution $h_{k_j}$ to (1.1) in $B(x_0,2r) \times (t_j,T)$ with lateral and initial boundary values $u_{k_j}$. We will do this via approximation in order to guarantee existence of solution to such a problem. Since $u_{k_j}$ is lower semicontinuous, there exists a sequence of continuous functions $\psi _{k_j,i}$, $i=1,2,\dots$, with $0 \le \psi _{k_j,i}\le \psi _{k_j,i+1} \le u_{k_j}$, $i=1,2,\dots$, and $\psi _{k_j,i} \nearrow u_{k_j}$ pointwise in $\Omega _T$ as $i\rightarrow \infty$. By Theorem 2.2, there exists a weak solution $h_{k_j,i}\in C(\overline{B(x_0,2r) \times (t_j,T)})$ to (1.1) in $B(x_0,2r) \times (t_j,T)$ with $h_{k_j,i}=\psi _{k_j,i}$ on $\partial _p(B(x_0,2r) \times (t_j,T))$. By the parabolic comparison principle in Theorem 2.3, we have $h_{k_j,i} \le u_{k_j}$ in $B(x_0,2r) \times (t_j,T)$ for every $i=1,2,\dots$. Lemma 4.4, with $s = t_j$ and $t < T$, implies

for every $\tau \in (t_j, t)$. On the other hand, we have

By combining the two inequalities above, we have

Passing to the limit $i\rightarrow \infty$ and using (4.3), we obtain

and thus

for every $\tau \in (t_j,t)$. By passing $j\rightarrow \infty$, we obtain

(4.4)

for every $\tau \in (t_0,t)$. Notice that the construction above can be done for any $r \in (0,r_0]$, which implies that (4.4) holds for every $\tau \in (t_0,t)$ and $r \in (0,r_0]$.

Let $(t_j)_{j\in {\mathbb {N}}}$ be a sequence for which (4.2) holds and let $r_j \rightarrow 0$ as $j\rightarrow \infty$. It follows that

(4.5)

This implies that

Let $\varepsilon \in [0, c_2(t-t_0))$ and, for every j large enough, choose the truncation levels $k_j$ so that

The constant $c_2 = c_2(n,p) \in (0,1)$ is from Lemma 4.3. Since $u_{k_j}$ is a nonnegative weak supersolution to (1.1) in $\Omega _T$, the weak Harnack inequality in Lemma 4.3 implies

for every $t \in [ t_j + \frac{3}{4}\varepsilon , t_j + \varepsilon ]$ and j large enough. Since sequence $r_j \rightarrow 0$ is arbitrary, the claim follows. $\square$

Next we give a characterization of the complementary class of the Barenblatt type supercaloric functions.

Theorem 4.6

Let $\frac{2n}{n+1}<p<2$ and let $\Omega$ be an open set in ${\mathbb {R}}^n$. Assume that u is a p-supercaloric function in $\Omega _T$. Then the following properties are equivalent:

• (i) $u\notin L^q_{{{\,\mathrm{loc}\,}}}(\Omega _T)$ for any $q> \frac{n}{p}(2-p)$,

• (ii) $u \notin L^{\frac{n}{p}(2-p)}_{{{\,\mathrm{loc}\,}}} (\Omega _T)$,

• (iii) there exists $\Omega '\Subset \Omega$ and $\delta \in (0, \frac{T}{2})$ such that

  $$\begin{aligned} {{\,\mathrm{ess\,sup}\,}}_{\delta<t<T-\delta } \int _{\Omega '} |u(x,t)|\, \, \mathrm {d}x=\infty , \end{aligned}$$

• (iii’) for every $\alpha \in (\frac{n}{p}(2-p), 1)$ there exists $\Omega '\Subset \Omega$ and $\delta \in (0, \frac{T}{2})$ such that

  $$\begin{aligned} {{\,\mathrm{ess\,sup}\,}}_{\delta<t<T-\delta } \int _{\Omega '} |u(x,t)|^\alpha \,\, \mathrm {d}x=\infty . \end{aligned}$$

• (iv) there exists $(x_0,t_0) \in \Omega _T$ such that

  $$\begin{aligned} \liminf _{\begin{array}{c} (x,t)\rightarrow (x_0,s) \\ t>s \end{array} } u(x,t)|x-x_0|^{\frac{p}{2-p}}>0 \end{aligned}$$

  for every $s>t_0$.

Proof

Since u is locally bounded from below, by adding a constant, we may assume that $u\ge 1$.

(ii) $\implies$ (i): Hölder’s inequality. (i) $\iff$ (iii’): Direct consequence of Theorem 3.10.

(iii’) $\implies$ (iii): Hölder’s inequality.

(iv) $\implies$ (iii’): There exists $r>0$ and $\delta >0$ such that

$$\begin{aligned} u(x,t) |x-x_0|^\frac{p}{2-p} \ge \varepsilon > 0 \end{aligned}$$

for every $(x,t) \in \left( B_r(x_0)\setminus \{x_0\} \right) \times (t_0, t_0+\delta )$. This implies (iii’).

(iv) $\implies$ (ii): The same argument as previous implication.

(iii) $\implies$ (iv): This is the implication which requires a bit more machinery, especially use of Harnack inequalities. From (iii) we have that there exists a point $t_0 \in (0,T)$ and a sequence $t_j \rightarrow t_0$ as $j \rightarrow \infty$ such that

$$\begin{aligned} \lim _{j\rightarrow \infty } \int _{\Omega '}u(x,t_j)\, \, \mathrm {d}x = \infty . \end{aligned}$$

The collection $\{B(x,r_x): x\in \overline{\Omega '},r_x>0\}$ is an open cover of $\overline{\Omega '}$. Since $\overline{\Omega '}$ is compact, there exists a finite subcover $\{B(x_i,r_i):i=1,\dots ,N\}$ and

$$\begin{aligned} \int _{\Omega '} u(x,t_j)\, \, \mathrm {d}x \le \sum _{i=1}^N \int _{B(x_i,r_i)} u(x,t_j)\, \, \mathrm {d}x \end{aligned}$$

for every $j=1,2,\dots$. By taking the limit $j \rightarrow \infty$ it follows that

$$\begin{aligned} \lim _{j \rightarrow \infty } \int _{B(x_i,r_i)} u(x,t_j) \, \, \mathrm {d}x = \infty \end{aligned}$$

(4.6)

for some $i\in \{1,\dots ,N\}$. We claim that there exists $x_0 \in B(x_i,r_i)$ such that

$$\begin{aligned} \lim _{j \rightarrow \infty } \int _{B(x_0,r)} u(x,t_j) \, \, \mathrm {d}x = \infty \end{aligned}$$

(4.7)

for arbitrarily small $r>0$. For a contradiction, assume that for every point $y\in B(x_i,r_i)$ there exists $r_y>0$ such that

$$\begin{aligned} \limsup _{j\rightarrow \infty } \int _{B(y,r)} u(x,t_j)\, \, \mathrm {d}x < \infty \end{aligned}$$

for every $0<r \le r_y$. The collection $\{B(y,r_y):y\in {\overline{B}}(x_i,r_i)\}$ is an open cover of ${\overline{B}}(x_i,r_i)$ and thus there exists a finite subcover $\{ B(y_k, r_k):k=1,2,\dots ,M\}$. Since

$$\begin{aligned} \int _{B(x_i,r_i)} u(x,t_j) \, \, \mathrm {d}x \le \sum _{k=1}^M \int _{B(y_k,r_k)} u(x,t_j) \, \, \mathrm {d}x, \end{aligned}$$

we obtain a contradiction by taking the limit $j \rightarrow \infty$ as the left hand side goes to infinity and right hand side does not. Thus (4.7) holds. Now the assumption for Lemma 4.5 holds which completes the proof. $\square$

Remark 4.7

We complete the proof of Theorem 3.10 by showing the remaining implication (ii) $\implies$ (i) by contraposition. Assume that the assertion (i) does not hold in Theorem 3.10. Then $u\notin L^q_{{{\,\mathrm{loc}\,}}}(\Omega _T)$ for any $q> \frac{n}{p}(2-p)$ and Theorem 4.6 implies that $u \notin L^{\frac{n}{p}(2-p)}_{{{\,\mathrm{loc}\,}}} (\Omega _T)$. Also the assertion $u \in L^\infty _{{{\,\mathrm{loc}\,}}} ( 0,T; L^1_{{{\,\mathrm{loc}\,}}}(\Omega ) )$ in Theorem 3.10 follows similarly from Theorem 4.6.