Existence of weak solutions for porous medium equation with a divergence type of drift term

Abstract

We consider degenerate porous medium equations with a divergence type of drift terms. We establish existence of nonnegative $L^{q}$-weak solutions (satisfying energy estimates or even further with moment and speed estimates in Wasserstein spaces), in case the drift term belongs to a sub-scaling (including scaling invariant) class depending on q and m caused by nonlinear structure of diffusion, which is a major difference compared to that of a linear case. It is noticeable that the classes of drift terms become wider, if the drift term is divergence-free. Similar conditions of gradients of drift terms are also provided to ensure the existence of such weak solutions. Uniqueness results follow under an additional condition on the gradients of the drift terms with the aid of methods developed in Wasserstein spaces. One of our main tools is so called the splitting method to construct a sequence of approximated solutions, which implies, by passing to the limit, the existence of weak solutions satisfying not only an energy inequality but also moment and speed estimates. One of crucial points in the construction is uniform Hölder continuity up to initial time for homogeneous porous medium equations, which seems to be of independent interest. As an application, we improve a regularity result for solutions of a repulsive Keller-Segel system of porous medium type.

Appendices

Appendix A Hölder continuity of homogeneous PME

Here we concern the homogeneous form of (1.1)

$$\begin{aligned} \partial _t u = \Delta u^m \ \text { in } \ Q_{T} \quad \text { for }\ m > 1, \end{aligned}$$

(A.1)

for $Q_{T}:= {\mathbb {R}}^d \times (0, T]$, $d\ge 2$, $0<T<\infty$ with a nonnegative initial data $u(x,0)=u_0 (x)$, $x\in {\mathbb {R}}^d$. In [11, 12], Caffarelli and Friedman proved the modulus of continuity of u in the entire half space, if $u_0 (x)$ is Hölder continuous. In [25, 26], DiBenedetto and Friedman determined Hölder exponent and constants quantitatively in the interior only depending upon the data. For $d=1$, the Hölder exponent is known explicitly as $\min \{1, \frac{1}{m-1}\}$ by Aronson [3, 4].

The Hölder continuity up to initial boundary is known, for exmaple, see [23, Section 6], as the Hölder exponent is depending upon the data and the modulus of continuity of the initial data. More specifically, for $u_0 (x) \in C^{\alpha _0}({\mathbb {R}}^d)$, a bounded nonnegative weak solution of (A.1) is uniformly Hölder continuous up to the initial boundary with the exponent $\alpha = \alpha (m, d,\alpha _0)\in (0,1)$ and the coefficient constant $c = c (m,d,\Vert u\Vert _{L^{\infty }(\bar{Q}_{T})})$. For more systemetic arguments, we refer [24, 27] for parabolic p-Laplace equation and for both p-Laplace and porous medium equations.

Although the uniform Hölder continuity of a bounded nonnegative weak solution of (A.1) is well-known, the dependency of the Hölder exponent with respect to the initial Hölder exponent seems not specified from previous works mentioned above. Therefore, our aim is to more clearly demonstrate the relation of interior and initial Hölder regularity. More precisely, in Theorem 5.2, we clarify the dependencies of uniform Hölder exponent as the minimum of the intertior Hölder exponent and given Hölder exponent of the initial data, that is, $\alpha = \min \{\alpha ^*, \alpha _0\}$ where $\alpha ^*= \alpha ^*(m,d)$ is determined from the interior Hölder result in Theorem A.1 and $\alpha _0 \in (0,1)$ is given from $u_0 \in {\mathcal {C}}^{\alpha _0}({{\mathbb {R}}}^d)$. To prove Theorem 5.2, we mainly modify arguments, the boundary Hölder continuity of parabolic p-Laplace equations up to the initial time, by DiBenedetto in [24, Theorem III]. Also typical arguments, for example, in [18, 23, 25, 34], are used to prove local Hölder continuity of (A.1) quantitatively.

Let $K_{r} (x_0)$ denote the $d-$dimensional cube centered at $x_0 \in {\mathbb {R}}^d$ and wedge 2r for $r>0$, that is,

$$\begin{aligned} K_{r}(x_0):= \{ x\in {\mathbb {R}}^d: \max _{1\le i \le d}|x^{i}-x_0^{i}| < r \}, \quad \text {for} \quad x=(x^1, \cdots , x^d). \end{aligned}$$

Besides let us denote two types of parabolic cyliners in $Q_{T}$, for $r>0$ and $0< t_0 < t_0 + r^2 \le T$,

$$\begin{aligned} Q_r:= K_r (x_0) \times (t_0, t_0 +r^2 ] \quad \text {and} \quad Q_{r}^{0}:= K_r (x_0) \times [0, r^2]. \end{aligned}$$

First, we introduce the interior Hölder continuity results in [23, 25,26,27]. When the divergence form of drift term is combined, we refer results in [18, 34].

Theorem A.1

Let u be a weak solution of (A.1) in $Q_{1} \subset Q_{T}$. Then there exist $\alpha ^*= \alpha ^*(m,d) \in (0,1)$ and $c = c (m, d, \Vert u\Vert _{L^{\infty }(Q_{1})})$, such that, u is uniformly Hölder continuous in $Q_{1/2}$ with the exponent $\alpha ^*$ and the coefficient c; that is, $\Vert u\Vert _{{{\mathcal {C}}}^{\alpha ^*} (Q_{1/2})} \le c$.

The main theorem is the following: the Hölder continuity up to the initial time.

Theorem A.2

Let u be a weak solution of (A.1) in $Q_{1}^{0}\subset Q_{T}$ with $u_0 \in {{\mathcal {C}}}^{\alpha _0} (K_{1})$. Then there exists ${\tilde{\alpha }} = \min \{\alpha _0, \alpha ^*\}$ where $\alpha ^*= \alpha ^*(m,d) \in (0,1)$ is given in Theorem A.1, and $c = c (m, d, \Vert u\Vert _{L^{\infty }(Q_{1}^{0})})$, such that, u is uniformly Hölder continuous in $Q_{1/2}^{0}$ with the exponent $\alpha$ and the coefficient c; that is, $\Vert u\Vert _{{{\mathcal {C}}}^{{\tilde{\alpha }}} (Q_{1/2}^0)} \le c$.

Theorem 5.2 is a direct consequence of Theorem A.1 and Theorem A.2.

The proof of Theorem 5.2: The global Hölder estimate up to $t=0$, is given by the standard covering arguments based on both interior and boundary estimates in Theorem A.1 and Theorem A.2. $\square$

It remains to prove Theorem A.2. In the first subsection, we provide energy and logarithmic type estimates up to the initial time which are fundamental to carry further arguments in the following subsection to prove Theorem A.2.

1.1 Energy estimates

In this section, we apply the idea in [24, Section II.4.iii] and local energy estimates in [18, Proposition 4.1 & 4.2] to obtain energy estimates of (5.2) up to the initial time.

We consider a weak solution of (A.1) that takes the initial data $u_0$ such that

$$\begin{aligned} \left( u_{h} (\cdot , 0) - u_0{}\right) _{\pm } \rightarrow 0, \quad \text {as } h\rightarrow 0 \quad \text {in } L^2_{{{\,\textrm{loc}\,}}} \text { sense,} \end{aligned}$$

(A.2)

where $u_{h}$ indicates the Stekelov average that

$$\begin{aligned} u_{h} (x, t) = \frac{1}{h} \int _{t}^{t+h} u(x, \tau ) \,d\tau , \quad \text {for all}\quad 0\le t \le T-h, \end{aligned}$$

which converges to u as $h\rightarrow 0$.

For some $k>0$, let us denote $(u - k)_{+} = \max \{ u-k, 0\}$ and $(u - k)_{-} = \max \{k-u, 0\}$.

Proposition A.3

Let u be a nonnegative bounded weak solution of (A.1). For $r>0$, $0<t<T$, suppose that $\zeta$ is a smooth cut-off function in $K_{r}(x_0)\times [0, t]$, which is $0 \le \zeta \le 1$, independent of t, and vanishing on the lateral boundary $\partial K_{r}(x_0)$. Let $\mu _{\pm }$ be positive constants such that

$$\begin{aligned} \mu _{+} \ge \mathop {\mathrm{ess\,sup}}\limits _{K_{r}(x_0)\times [0, t]} u \quad \text {and} \quad \mu _{-} \le \mathop {\mathrm{ess\,inf}}\limits _{K_{r}(x_0)\times [0, t]} u. \end{aligned}$$

Moreover, for every level $k > 0$, suppose that

$$\begin{aligned} {\left\{ \begin{array}{ll} \mu _{+} - k > \mathop {\mathrm{ess\,sup}}\nolimits _{K_{r} (x_0)} u_0, &{} \quad \text {for}\quad (u - \mu _{+} - k)_{+}, \\ \mu _{-} + k < \mathop {\mathrm{ess\,inf}}\nolimits _{K_{r} (x_0)} u_0, &{} \quad \text {for}\quad (u - \mu _{-} + k)_{-}. \end{array}\right. } \end{aligned}$$

(A.3)

Then the following estimate holds

$$\begin{aligned} \begin{aligned}&\sup _{0 \le \tau \le t} \int _{K_r (x_0)} \left( u (\cdot , \tau ) - \mu _{\pm } \pm k\right) _{\pm }^{2} \zeta ^2 \,dx + m \int _{0}^{t} \int _{K_{r}(x_0)} u^{m-1} \left| \nabla \left( u - \mu _{\pm } \pm k\right) _{\pm }\right| ^2 \zeta ^2 \,dx\,d\tau \\&\quad \le 16m \int _{0}^{t} \int _{K_{r}(x_0)} u^{m-1} \left( u - \mu _{\pm } \pm k\right) _{\pm }^{2} |\nabla \zeta |^2 \,dx\,d\tau . \end{aligned} \end{aligned}$$

(A.4)

Moreover, it holds that, for any $\tau \in (0, t]$,

$$\begin{aligned} \begin{aligned}&\int _{K_r (x_0)} \Psi _{\pm }^{2}(u (\cdot , \tau )) \zeta ^2 \,dx + 2m \int _{0}^{t} \int _{K_{r}(x_0)} u^{m-1} |\nabla u|^2 \Psi _{\pm } \left| \Psi '_{\pm }(u)\right| ^2 \zeta ^2 \,dx\,d\tau \\&\quad \le 16 m \int _{0}^{t} \int _{K_{r}(x_0)} u^{m-1} \Psi _{\pm }^{2}(u) |\nabla \zeta |^2 \,dx\,d\tau , \end{aligned} \end{aligned}$$

(A.5)

where, for $\delta \in (0, 1)$,

$$\begin{aligned} \Psi _{\pm } (u) = \ln ^{+} \left[ \frac{k}{(1+\delta )k - \left( u - \mu _{\pm } \pm k\right) _{\pm }}\right] . \end{aligned}$$

Proof

A weak solution formulation of (A.1) rephrased in terms of Stekelov averages is given

$$\begin{aligned} \int _{0}^{t} \int _{K_{r}(x_0)} \left[ \partial _t u_h \varphi + \nabla u_h^{m} \nabla \varphi \right] \,dx\,dt = 0, \end{aligned}$$

for all $0<t<T-h$ and for all $\varphi \in W_{o}^{1,p} (\Omega ) \cap L^{\infty }_{x}(\Omega )$, $\varphi \ge 0$ (see [24, Section II.2.i]).

To obtain (A.4), we test $\varphi = \pm 2\left( u_{h} - \mu _{\pm } \pm k\right) _{\pm } \zeta ^2$. Then

$$\begin{aligned} \begin{aligned} \int _{0}^{t} \int _{K_{r}(x_0)} \partial _t u_h \varphi \,dx\,dt&= \int _{0}^{t} \int _{K_{r}(x_0)} \partial _t \left[ \left( u_{h} - \mu _{\pm } \pm k\right) _{\pm } \zeta \right] ^2 \,dx\,dt \\&= \int _{K_{r}(x_0)\times \{t\}} \left( u_{h} - \mu _{\pm } \pm k\right) _{\pm }^2 \zeta ^2 \,dx\,dt \\&\quad - \int _{K_{r}(x_0)} \left( u_{h} (\cdot , 0) - \mu _{\pm } \pm k\right) _{\pm }^2 \zeta ^2 \,dx\,dt, \end{aligned} \end{aligned}$$

in which the second term vanishes as $h \rightarrow 0$ because of (A.2) and (A.3). We skip the computation of $\iint \nabla u_h^{m} \nabla \varphi \,dx\,dt$ because it is similar to [18, Proposition 4.1]. The combination of above estimates yields (A.4).

Now to obtain (A.5), we note that, from the definition of $\Psi _{\pm } (u)$, it follows that

$$\begin{aligned} \Psi _{\pm }(u) = 0 \quad \text {whenever } (u- \mu _{\pm }\pm k)_{\pm } = 0. \end{aligned}$$

Then (A.3) provides that

$$\begin{aligned} \int _{K_{r} (x_0)} \Psi ^{2} (u_h) (x, 0) \zeta ^2 \,dx \rightarrow 0, \quad \text {as } h\rightarrow 0. \end{aligned}$$

Therefore, (A.5) follows by carrying almost the same computations as in [18, Proposition 4.2]. $\square$

1.2 Proof of Theorem A.2.

Here, we prove Theorem A.2 by modifying methodologies in [24, Section III]. Because of translation invariant property, we assume $x_0 = 0$ without loss of generality. First, we construct a parabolic cyliner

$$\begin{aligned} Q_{2R, R^{2-\epsilon }} = K_{2R}(0)\times [0, R^{2-\epsilon }], \quad \text {for}\quad R>0, \end{aligned}$$

(A.6)

where $\epsilon = \epsilon (m)$ is a positive number determined later. Moreover, let us set

$$\begin{aligned} \mu _{+}:= \mathop {\mathrm{ess\,sup}}\limits _{Q_{2R, R^{2-\epsilon }}} u, \qquad \mu _{-}:= \mathop {\mathrm{ess\,inf}}\limits _{Q_{2R, R^{2-\epsilon }}} u, \quad \text {and} \quad \omega := \mathop {\mathrm{ess\,osc}}\limits _{Q_{2R, R^{2-\epsilon }}} u = \mu _{+} - \mu _{-}. \end{aligned}$$

(A.7)

Now we construct the intrinsic parabolic cylinder such that

$$\begin{aligned} Q_{R, a_0 R^{2}}:= K_{R}(0) \times [0, a_0 R^2], \quad \text {where} \quad \frac{1}{a_0}=\left( \frac{\omega }{4}\right) ^{m-1}. \end{aligned}$$

These parabolic boxes are lying on the bottom of ${\mathbb {R}}^d\times {\mathbb {R}}_{+}$.

We establish the decay of oscillation in the next proposition, which is similar to [24, Proposition III.11.1].

Proposition A.4

There exist constants $\epsilon _0 = \epsilon _{0} (m,d,\epsilon ) \in (0,1)$ for $\epsilon$ given in (A.6), $\eta _1 = \eta _1 (m,d)\in (0,1)$, and $\eta _2 = \eta _2 (m,d) \in (0,1)$ which satisfy the following. Let us construct the sequences with $R_0 = R$ and $\omega _0 = \omega$:

$$\begin{aligned} R_n = {\eta _2}^{n} R, \quad \omega _{n+1} = \max \{ \eta _1 \omega _n, \, 4 R_n^{\epsilon _0}\}, \quad n=1,2,\ldots . \end{aligned}$$

(A.8)

Also, construct the family of intrinsic parabolic cylinders:

$$\begin{aligned} Q_n:= K_{R_n} \times [0, \left( \frac{\omega _{n}}{4}\right) ^{1-m} R_n^2], \quad n=1,2,\ldots . \end{aligned}$$

that satisfies $Q_{n+1} \subset Q_{n}$. Then the following holds: for all $n=0,1,2,\ldots$,

$$\begin{aligned} \mathop {\mathrm{ess\,osc}}\limits _{Q_n} u \le \max \{\omega _n, \, 2 \mathop {\mathrm{ess\,osc}}\limits _{K_{R_n}} u_0\}. \end{aligned}$$

(A.9)

Proof

For any $R>0$, let us start from $Q_0:= Q_{2R, R^{2-\epsilon }}$ in (A.6) where $\epsilon \in (0,1)$ to be determined later. For $\mu _{\pm }$ and $\omega$ given in (A.7), we construct intrinsically scaled parabolic cylinders

$$\begin{aligned} Q_{R}^{\omega }:= K_{R}(0) \times \left[ 0, \left( \frac{\omega }{4}\right) ^{1-m} R^2\right] . \end{aligned}$$

Then there are two cases: either

$$\begin{aligned} Q_{R}^{\omega } \subset Q_0 \quad \text {which gives} \quad \omega > 4 R^{\frac{\epsilon }{m-1}}, \end{aligned}$$

(A.10)

or

$$\begin{aligned} Q_0 \subset Q_{R}^{\omega } \quad \text {which gives} \quad \omega \le 4 R^{\frac{\epsilon }{m-1}}. \end{aligned}$$

(A.11)

If (A.11) holds, then (A.8) follows directly with $\epsilon _0 = \frac{\epsilon }{m-1}$.

Now let us assume (A.10) and set

$$\begin{aligned} \mu ^{+}_{0} = \mathop {\mathrm{ess\,sup}}\limits _{K_R} u_0, \quad \mu ^{-}_{0}= \mathop {\mathrm{ess\,inf}}\limits _{K_R} u_0, \quad \omega _0 = \mathop {\mathrm{ess\,osc}}\limits _{K_R} u_0 = \mu ^{+}_{0} - \mu ^{-}_{0}. \end{aligned}$$

Then there are two inequalities to cosider:

$$\begin{aligned} \mu ^{+} - \frac{\omega }{4} \le \mu ^{+}_{0} \quad \text {and} \quad \mu ^{-} + \frac{\omega }{4} \ge \mu ^{-}_{0}. \end{aligned}$$

(A.12)

If both inequalities in (A.12) hold, then the subtraction of the second from the first gives

$$\begin{aligned} \mathop {\mathrm{ess\,osc}}\limits _{Q_{R}^{\omega }} u \le 2 \mathop {\mathrm{ess\,osc}}\limits _{K_R} u_0, \end{aligned}$$

which holds (A.9) and there is nothing to prove.

If the second inequality in (A.12) is violated, then we choose $k =\frac{\omega }{4}$ and it gives

$$\begin{aligned} \mu ^{-} + k < \mu _{0}^{-}, \end{aligned}$$

which satisfies (A.3). Therefore, we have energy estimates (A.4) and (A.5) for $(u - \mu ^{-} - k)_{-}$. With these estimates, we are able to carry the quantitative method such as the expansion of positivity and DeGiorgi iteration as in [18]. We obtain that there exists $\delta _1 = \delta _1 (m,d) \in (0,1)$ such that

$$\begin{aligned} \mathop {\mathrm{ess\,inf}}\limits _{Q_{R/8}^{\omega /4}}u \ge \mu _{-} + \delta _1 \omega , \quad \text {for} \quad Q_{R/8}^{\omega /4}:= K_{R/8}\times \left[ 0, \left( \frac{\omega }{4}\right) ^{1-m}\left( \frac{R}{8}\right) ^2\right] . \end{aligned}$$

If the first inequality in (A.12) fails, then we have (A.4) and (A.5) for $(u - \mu ^{+}+k)_{+}$ because (A.3) holds $k=\frac{\omega }{4}$. Then again the same arguments in [18] deduce that there exists $\delta _2 = \delta _2 (m,d) \in (0,1)$ such that

$$\begin{aligned} \mathop {\mathrm{ess\,sup}}\limits _{Q_{R/8}^{\omega /4}}u \le \mu _{+} - \delta _2 \omega . \end{aligned}$$

From above, when (A.12) fails, we choose $\eta _1 = \max \{ 1 - \delta _1, \ 1-\delta _2\}$ to obtain

$$\begin{aligned} \mathop {\mathrm{ess\,osc}}\limits _{Q_{R/8}^{\omega /4}} u \le \eta _1 \omega , \end{aligned}$$

which parallels to Lemma III.11.1 in [24].

Finally, we construct nested and shrinking family of parabolic cylinders. First, let us set

$$\begin{aligned} {\eta _2} \le \frac{1}{8} \, {\eta _1}^{\frac{m-1}{2}}, \quad \text {for} \quad {\eta _1} = \max \{ 1 - \delta _1, \ 1-\delta _2\} \in (0,1). \end{aligned}$$

(A.13)

This provides

$$\begin{aligned} Q_1:=Q_{{\eta _2} R}^{{\eta _1} \omega / 4} \subset Q_{R/8}^{\omega /4} \subset Q_{R}^{\omega /4} \subset Q_{R}^{\omega /4}=: Q_0. \end{aligned}$$

Then by repeating the same iteration, we complete the proof. $\square$

From Proposition A.4, we derive the Hölder continuity in the next lemma.

Lemma A.5

Let u be a nonnegative weak solution of (A.1) with $u_0 \in {{\mathcal {C}}}^{\alpha _0}({\mathbb {R}}^d)$. Then there exist $\epsilon _0 = \epsilon _0 (m) \in (0,1)$, $\alpha _i = \alpha _i (m,d) \in (0,1)$ for $i=1,2$, and $c= c(m,d) >1$, such that, for any $R>0$ and $0< r \le R$, it holds

$$\begin{aligned} \mathop {\mathrm{ess\,osc}}\limits _{Q_r^0} u \le c \left( \omega + R^{\epsilon _0} + R^{\alpha _0}\right) \left( \frac{r + (\omega /4)^{\frac{m-1}{2}} r}{R}\right) ^{{\tilde{\alpha }}}, \end{aligned}$$

for ${\tilde{\alpha }} = \min \{\alpha _0, \alpha _1, \alpha _2, \epsilon _0/2 \}$.

Proof

For ${\eta _1}, {\eta _2} \in (0,1)$ given in Proposition A.4, let us set $R_n = {\eta _2}^n R$ and $\omega _n = {\eta _1}^n \omega$. For any $0 < \rho \le R$, there exist nonnegative integers k and l such that

$$\begin{aligned} R_{k+1} < r \le R_{k} \end{aligned}$$

(A.14)

and

$$\begin{aligned} \left( \frac{\omega _{l+1}}{4}\right) ^{1-m}R_{l+1}^{2} < r^2 \le \left( \frac{\omega _{l}}{4}\right) ^{1-m}R_{l}^{2}, \end{aligned}$$

(A.15)

which means $Q_{r}$ belongs to either $Q_k$ or $Q_l$. Therefore, by Proposition A.4, it implies

$$\begin{aligned} \mathop {\mathrm{ess\,osc}}\limits _{Q_r} u \le \max \{ \omega _{k}, \ \omega _{l}\}, \end{aligned}$$

and, moreover, it gives (because $\mathop {\mathrm{ess\,osc}}\nolimits _{K_{R_n}} u_0 \le R_n^{\alpha _0}$ by the Hölder continuity of $u_0$)

$$\begin{aligned} \omega _{n+1} \le {\eta _1} \omega _n + 4R_n^{\epsilon _0} + 2 R_n^{\alpha _0}. \end{aligned}$$

First, we deduce from the iteration that

$$\begin{aligned} \omega _n \le {\eta _1}^n \omega + 4 \left( \sum _{i=1}^{n}{\eta _1}^{n-i}{\eta _2}^{\epsilon _0 (i-1)}\right) R^{\epsilon _0} + 2\left( \sum _{i=1}^{n}{\eta _1}^{n-i}{\eta _2}^{\alpha _0 (i-1)}\right) R^{\alpha _0} = I_{n} + II_{n} + III_{n}. \end{aligned}$$

Then we consider two cases separately.

If (A.14) holds, then we deduce the following, from the LHS of (A.14),

$$\begin{aligned} {\eta _2}^{k+1}< \frac{r}{R} \quad \Longrightarrow \quad k+1 < \log _{{\eta _2}}r/R = \log _{{\eta _1}} (r/R)^{\alpha _1}, \quad \text {for} \quad \alpha _1 = \left| \log _{{\eta _2}} {\eta _1} \right| . \end{aligned}$$

Therefore, we easily obtain

$$\begin{aligned} {\eta _1}^{k} < {\eta _1}^{-1}\left( \frac{r}{R}\right) ^{\alpha _1} \quad \Longrightarrow \quad I_{k} \le {\eta _1}^{-1}\omega \left( \frac{r}{R}\right) ^{\alpha _1}. \end{aligned}$$

(A.16)

Recalling (A.13), without loss of generality, let us fix

$$\begin{aligned} {\eta _2} = {\eta _1}^{m-1} < \frac{1}{8}{\eta _1}^{\frac{m-1}{2}} \quad \Longrightarrow \quad {\eta _1} = {\eta _2}^{\frac{1}{m-1}}. \end{aligned}$$

(A.17)

Then now we observe that

$$\begin{aligned} II_{k}= & {} 4 \left( \sum _{i=1}^{k}{\eta _1}^{k-i}{\eta _2}^{\epsilon _0 (i-1)}\right) R^{\epsilon _0} = 4 \left( \sum _{i=1}^{k}{\eta _2}^{\frac{1}{m-1}(k-i)}{\eta _2}^{\epsilon _0 (i-1)}\right) R^{\epsilon _0} \\\le & {} 4 \left( \sum _{i=1}^{k}{\eta _2}^{\epsilon _0 (k-1)}\right) R^{\epsilon _0}, \end{aligned}$$

by choosing $\epsilon _0 = \min \{ 1/2, \ 1/(m-1) \}$. Therefore, it yields

$$\begin{aligned} II_{k} \le 4 {\eta _2}^{-\epsilon _0} k ({\eta _2}^{k} R )^{\epsilon _0} \le 4 {\eta _2}^{-\epsilon _0} \left( \frac{R}{r}\right) ^{\epsilon _0 /2} \left( \frac{r}{{\eta _2}}\right) ^{\epsilon _0}= 4{\eta _2}^{-2\epsilon _0} R^{\epsilon _0} \left( \frac{r}{R}\right) ^{\epsilon _0 / 2}, \end{aligned}$$

(A.18)

by (A.14) and by choosing $k \in {\mathbb {N}}$ such that

$$\begin{aligned} k < {\eta _2}^{-k \epsilon _0 /2} \le \left( \frac{R}{r}\right) ^{\epsilon _0 /2}. \end{aligned}$$

(A.19)

The second inequality of (A.19) comes form the RHS of (A.14), and the first inequality of (A.19) is true in general by choosing $\frac{1}{{\eta _2}}$ large enough. Let us fix

$$\begin{aligned} {\tilde{\alpha }} = \min \{ \alpha _0, \alpha _1, \epsilon _0 / 2\}, \end{aligned}$$

(A.20)

and then it follows

$$\begin{aligned} \begin{aligned} III_{k}&= 2\left( \sum _{i=1}^{k}{\eta _1}^{k-i}{\eta _2}^{\alpha _0 (i-1)}\right) R^{\alpha _0} \le 2\left( \sum _{i=1}^{k}{\eta _2}^{\epsilon _0 (k-i)}{\eta _2}^{\alpha _0 (i-1)}\right) R^{\alpha _0} \\&\le 2 {\eta _2}^{-\epsilon _0} \left( \sum _{i=1}^{k}{\eta _2}^{(\epsilon _0 -{\tilde{\alpha }}) (k-i +1)}\right) ({\eta _2}^{k} R)^{{\tilde{\alpha }}} < \frac{2 {\eta _2}^{-2\epsilon _0}}{{\eta _2}^{-\epsilon _0 /2} - 1} R^{\alpha _0} \left( \frac{r}{R}\right) ^{{\tilde{\alpha }}}, \end{aligned} \end{aligned}$$

(A.21)

because, by the LHS of (A.14),

$$\begin{aligned} \sum _{i=1}^{k}{\eta _2}^{(\epsilon _0 -{\tilde{\alpha }}) (k-i +1)}< \frac{1}{{\eta _2}^{\alpha _0 - \epsilon _0} - 1} \le \frac{1}{{\eta _2}^{- \epsilon _0 /2} - 1}, \quad \text {and}\quad {\eta _2}^{-{\tilde{\alpha }}} < {\eta _2}^{-\epsilon _0}. \end{aligned}$$

Finally, the summation of (A.16), (A.18), (A.21), and the choice of $\alpha _0$ in (A.20) imply that

$$\begin{aligned} \omega _{l}\le & {} {\eta _1}^{-1}\omega \left( \frac{r}{R}\right) ^{\alpha _1} + 4{\eta _2}^{-2\epsilon _0} R^{\epsilon _0} \left( \frac{r}{R}\right) ^{\epsilon _0 / 2} + \frac{2 {\eta _2}^{-2\epsilon _0}}{{\eta _2}^{-\epsilon _0 /2} - 1} R^{\alpha _0} \left( \frac{r}{R}\right) ^{{\tilde{\alpha }}} \\\le & {} c \left( \omega + R^{\epsilon _0} + R^{\alpha _0}\right) \left( \frac{r}{R}\right) ^{{\tilde{\alpha }}}. \end{aligned}$$

If (A.15) hold, then let us rewrite (A.15) in the following:

$$\begin{aligned} \left( {\eta _1}^{1-m} {\eta _2}^2 \right) ^{l+1} R^2 < \left( \frac{\omega }{4}\right) ^{m-1} r^2 \ (=:\tilde{r}^2) \le \left( {\eta _1}^{1-m} {\eta _2}^2 \right) ^{l} R^2. \end{aligned}$$

(A.22)

Let us set

$$\begin{aligned} \tilde{{\eta _2}} = {\eta _1}^{\frac{1-m}{2}}{\eta _2}, \quad \text {and} \quad \tilde{r}^2 = \left( \frac{\omega }{4}\right) ^{m-1} r^2. \end{aligned}$$

Then (A.22) is now simplified as

$$\begin{aligned} \tilde{{\eta _2}}^{l+1} R < \tilde{r} \le \tilde{{\eta _2}}^l R. \end{aligned}$$

Without loss of generality, assume that $\tilde{r} \le R$. Moreover, the choice of ${\eta _2}$ as in (A.17) gives

$$\begin{aligned} {\eta _1}^{1-m} {\eta _2}^{2} = {\eta _1}^{m-1} = {\eta _2} \quad \text {and} \quad \tilde{{\eta _2}} = {\eta _2}^{3/2}. \end{aligned}$$

Therefore, the first inequality in (A.22) yields

$$\begin{aligned} l+1 < \log _{{\eta _1}} \left( \tilde{r}/R\right) ^{\alpha _2} \quad \Longrightarrow \quad I_l \le {\eta _1}^{-1} \omega \left( \frac{\tilde{r}}{R}\right) ^{\alpha _2}, \quad \text {for}\quad \alpha _2 = \left| \log _{\tilde{{\eta _2}}} {\eta _1} \right| . \end{aligned}$$

Then we carry similar analysis for $II_l$ and $III_l$ as handling $II_k$ in (A.18) and $III_k$ as in (A.21) with ${\tilde{\alpha }} = \min \{\alpha _0, \alpha _2, \frac{\epsilon _0}{2}\}$ to have

$$\begin{aligned} \omega _{l}\le & {} {\eta _1}^{-1} \omega \left( \frac{\tilde{r}}{R}\right) ^{\alpha _2}+ 4 \tilde{{\eta _2}}^{-2\epsilon _0} R^{\epsilon _0}\left( \frac{\tilde{r}}{R}\right) ^{\epsilon _0 /2}+ \frac{2 {\eta _2}^{-2\epsilon _0}}{{\eta _2}^{-\epsilon _0 /2} - 1} R^{\alpha _0}\left( \frac{\tilde{r}}{R} \right) ^{{\tilde{\alpha }}}\\\le & {} c \left( \omega + R^{\epsilon _0} + R^{\alpha _0}\right) \left( \frac{\tilde{r}}{R} \right) ^{{\tilde{\alpha }}}. \end{aligned}$$

Therefore, we complete the proof by choosing ${\tilde{\alpha }} = \min \{\alpha _0, \alpha _1, \alpha _2, \frac{\epsilon _0}{2} \}$ and by combining all estimates above. $\square$

Once we have Lemma A.5, Theorem A.2 follows from standard computations (refer [18, 24, 34]).

Appendix B Figure supplements

• Figures 11 and 12 (Theorem 2.9for case $1<m\le 2$, $q>1$): Refer Remark 2.10 (iii) and Fig. 6. As d increases, the point ${\textbf {b}}$ approaches closer to the origin and ${\textbf {A}}$ may locate on the right hand side of ${\textbf {B}}$ and ${\textbf {b}}$.

Fig. 11

$\max \{2,\frac{2m}{(2m-1)(m-1)}\} < d \le \frac{2m}{m-1}$

Fig. 12

$d > \frac{2m}{m-1}$

• Figure 13 (Theorem 2.9for case $m>2$, $q\ge m-1$): Refer Remark 2.10 (iv) and Fig. 7.

Fig. 13

$m>2$, $q\ge m-1$, $d >\frac{2m}{m-1}$

Fig. 14

$m>2$, $q> \frac{m}{2}$, $d>2$

• Figure 14 (Theorem 2.11for case $m>2$, $q > \frac{m}{2}$): Refer Remark 2.12 (iii) and Fig. 8.

  When $d=2$, the line $\overline{{\textbf {CH}}}$ is excluded from the region ${\mathcal {R}}({\textbf {ABCHI}})$.

• Figures  15 and 16 (Theorem 2.13for case $1<m\le 2$, $q>1$): Refer Remark 2.14 (ii) and Fig. 9.

Fig. 15

$\max \{ 2, \frac{2m}{(2m-1)(m-1)}\} < d \le \frac{2m}{m-1}$

Fig. 16

$d > \frac{2m}{m-1}$

• Figures 17 and 18 (Theorem 2.13for case $m>2$, $q\ge \frac{m}{2}$): Refer Remark 2.14 (ii) and Fig. 9. We note that $\frac{2m}{(2m-1)(m-1)} < 2$ for $m>2$, thus it is enough to consider two cases.

Fig. 17

$m>2$, $q\ge \frac{m}{2}$, $2< d \le \frac{2m}{m-1}$

Fig. 18

$m > 2$, $q \ge \frac{m}{2}$, $d > \frac{2\,m}{m-1}$

Fig. 19

$m> 2$, $q \ge \frac{m}{2}$

• Figure 19 (Theorem 2.15and 2.16for case $m>2$, $q> \frac{m}{2}$): Refer Remark 2.17 (ii) and Fig. 10.

1.1 Embedding

The following figures illustrate strategies of applying embedding arguments in the temporal varaiable in the second part os Theorem 2.9, 2.13, and 2.16. Unfortunately, embedding arguments in spatial variables do not work because ${{\mathbb {R}}}^d$ is unbounded. If spatial embedding arguments are applicable, then, for example in Fig. 20, it gives a way to include the region ${\mathcal {R}}({\textbf {aAF}})$ by searching $(q_1^*, q_2)$ in ${\mathcal {R}}({\textbf {ABCDEF}})$ by decreasing $q_1^*< q_1$ for some $q^*\in (1, q)$. We are preparing a parallel paper constructing the existence results of (1.1) in $\Omega \times (0, T]$ for $\Omega$ bounded in ${{\mathbb {R}}}^{d}$.

• Figure 20 (Theorem 2.9(ii) for case $1<m\le 2$, $q>1$): Starting a pair $(q_1, q_2)$ in ${\mathcal {R}}({\textbf {bCB}})$, one can decrease $q_2$ to $q_{2}^*$ until $(q_1, q_{2}^{*})$ hits the line $\overline{{\textbf {BC}}}$ or any pair belonging ${\mathcal {R}}({\textbf {ABCDEF}})$. Then there exists $q^*\in (1, q)$ in which $(q_1, q_{2}^{*})$ lies on ${\mathcal {S}}_{m,q^*}^{(q_1, q_2^*)}$. Then we apply the existence results in Theorem 2.9$\textit{(i)}$ for $1 < p \le {\lambda _{q^*}}$. When $\max \{2,\frac{2\,m}{(2\,m-1)(m-1)}\} < d \le \frac{2\,m}{m-1}$ or $d > \frac{2\,m}{m-1}$, we repeat the same strategy with Fig. 11, 12, respectively.

Fig. 20

$1<m\le 2$, $q>1$, $2<d \le \max \{2, \frac{2m}{(2m-1)(m-1)}\}$

• Figure 21 (Theorem 2.9(ii) for case $m > 2$, $q\ge m-1$: Refer Fig. 7. The same strategy works to Fig. 13.

Fig. 21

$m>2$, $q\ge m-1$, $2 < d \le \frac{2m}{m-1}$

Fig. 22

$1<m\le 2$, $q>1$, $2 < d \le \max \{2,\frac{2m}{(2m-1)(m-1)}\}$

Fig. 23

$m>2$, $q\ge \frac{m}{2}$, $2< d \le \frac{2m}{m-1}$

Fig. 24

$1<m\le 2$, $q>1$

• Figure 22 (Theorem 2.13(ii) for case $1<m\le 2$, $q>1$): With Fig. 9, we explain strategy of searching $q_2^*$ and $q^*$. There are two regions ${\mathcal {R}} ( {\textbf {aAFE}})$ if $\frac{2\,m}{m-1}< q_1 < \infty$ and ${\mathcal {R}} ( {\textbf {BbhDC}})$ if $\frac{md}{1+d(m-1)}< q_1 < d(2\,m-1)$, in which we find $q_2^*$ such that $(q_1, q_2^*)$ belongs to ${\mathcal {R}} ({\textbf {ABCDEF}})$. Then we are able to search matching $q^*\in (1, q]$ and able to apply Theorem 2.13 (i). Also, we can apply the same strategy with Fig. 15, 16.

• Figure 23 (Theorem 2.13(ii) for case $m > 2$, $q\ge \frac{m}{2}$): With Fig. 17, we explain strategy of searching $q_2^*$ and $q^*$. There are two regions ${\mathcal {R}} ( {\textbf {jAFI}})$ and ${\mathcal {R}} ( {\textbf {GAbhHCG}})$, in which we find $q_2^*$ such that $(q_1, q_2^*)$ belongs to ${\mathcal {R}} ({\textbf {FGCHI}})$ to apply Theorem 2.13 (i). Also, we can apply the same strategy with Fig. 18.

• Figure 24 (Theorem 2.16(ii)): The following figure visualizes the idea of searching $q^*$ and $\tilde{q}^*_2$ with Fig. 10. Also we can repeat the same strategy with Fig. 19.