Pointwise Bounds and Blow-up for Systems of Nonlinear Fractional Parabolic Inequalities

TL;DR

This paper derives optimal pointwise bounds for nonnegative solutions of a nonlinear fractional parabolic system, analyzing their behavior near initial time and at infinity, and establishing conditions for blow-up or boundedness.

Contribution

It introduces new bounds for solutions of fractional parabolic inequalities, extending classical results to fractional operators and systems with specific initial conditions.

Findings

Established optimal bounds as t→0+

Derived bounds as t→∞

Identified conditions for solution blow-up or boundedness

Abstract

We investigate nonnegative solutions $u(x,t)$ and $v(x,t)$ of the nonlinear system of inequalities \[0\leq(\partial_t -\Delta)^\alpha u\leq v^\lambda\] \[ 0\leq (\partial_t -\Delta)^\beta v\leq u^\sigma\] in $\mathbb{R}^n \times\mathbb{R}$, $n\geq 1$, satisfying the initial conditions \[ u=v=0\quad\text{ in }\mathbb{R}^n \times(-\infty,0) \] where $\lambda,\sigma,\alpha$, and $\beta$ are positive constants. Specifically, using the definition of the fractional heat operator $(\partial_t-\Delta)^\alpha$ given in \cite{T}, we obtain, when they exist, optimal pointwise upper bounds on $\mathbb{R}^n \times(0,\infty)$ for nonnegative solutions $u$ and $v$ of this initial value problem with particular emphasis on these bounds as $t\to0^+$ and as $t\to\infty$.

Equations

$$\begin{array}{l}0\le ({\partial }_t-\Delta {)}^{\alpha }u\le v^{\lambda }\\ 0\le ({\partial }_t-\Delta {)}^{\beta }v\le u^{\sigma }\end{array}\}\text{ in }{\mathbb{R}}^n\times \mathbb{R},n\ge 1$$

$$u=v=0\text{ in }{\mathbb{R}}^n\times (-\infty ,0)$$

$$\begin{array}{l}0\le ({\partial }_t-\Delta {)}^{\alpha }u\le v^{\lambda }\\ 0\le ({\partial }_t-\Delta {)}^{\beta }v\le u^{\sigma }\end{array}\}\text{ in }{\mathbb{R}}^n\times \mathbb{R},n\ge 1$$

$$u=v=0\text{ in }{\mathbb{R}}^n\times (-\infty ,0),$$

$$0\le ({\partial }_t-\Delta {)}^{\alpha }u\le u^{\lambda }\text{ in }{\mathbb{R}}^n\times \mathbb{R}$$

$$u=0\text{ in }{\mathbb{R}}^n\times (-\infty ,0),$$

$$({\partial }_t-\Delta {)}^{\alpha }:Y_{\alpha }^p\to X^p$$

$$\left(p>1\text{ and }0<\alpha <\frac{n+2}{2p}\right)\text{ or }\left(p=1\text{ and }0<\alpha \le \frac{n+2}{2p}\right)$$

$$J_{\alpha }:X^p\to Y_{\alpha }^p$$

$$X^p:=\bigcap _{T\in \mathbb{R}}{L}^p({\mathbb{R}}^n\times {\mathbb{R}}_T),{\mathbb{R}}_T:=(-\infty ,T),$$

$$J_{\alpha }f(x,t):={\iint }_{{\mathbb{R}}^n\times {\mathbb{R}}_t}{\Phi }_{\alpha }(x-\xi ,t-\tau )f(\xi ,\tau )d\xi d\tau$$

$$Y_{\alpha }^p:=J_{\alpha }(X^p).$$

$$\Vert f{\Vert }_{L^p({\mathbb{R}}^n\times {\mathbb{R}}_T)}<\infty \text{ for all }T\in \mathbb{R}.$$

$${\Phi }_{\alpha }(x,t):=\frac{t^{\alpha -1}}{\Gamma (\alpha )}\frac{1}{(4\pi t{)}^{n/2}}{e}^{-|x{|}^2/(4t)}\chi (t)$$

$$({\partial }_t-\Delta {)}^{\alpha }u=f,$$

$$u\in Y_{\alpha }^p\text{ and }v\in Y_{\beta }^q$$

$$\begin{array}{l}0\le ({\partial }_t-\Delta {)}^{\alpha }u\le v^{\lambda }\\ 0\le ({\partial }_t-\Delta {)}^{\beta }v\le u^{\sigma }\end{array}\}\text{ in }{\mathbb{R}}^n\times \mathbb{R},n\ge 1$$

$$u=v=0\text{ in }{\mathbb{R}}^n\times (-\infty ,0),$$

$$p,q\in [1,\infty ),\lambda ,\sigma ,\alpha ,\beta \in (0,\infty ),$$

$$\left(p>1\text{ and }0<\alpha <\frac{n+2}{2p}\right)\text{ or }\left(p=1\text{ and }0<\alpha \le \frac{n+2}{2p}\right)$$

$$\left(q>1\text{ and }0<\beta <\frac{n+2}{2q}\right)\text{ or }\left(q=1\text{ and }0<\beta \le \frac{n+2}{2q}\right).$$

$$u=J_{\alpha }(({\partial }_t-\Delta {)}^{\alpha }u)\ge 0\text{ in }{\mathbb{R}}^n\times \mathbb{R}$$

$$(n+2-2p\alpha )q^2\sigma \le (n+2-2q\beta )p^2\lambda ,$$

$$2p\alpha <n+2$$

$$2p\alpha =n+2.$$

$$2q\beta <n+2,$$

$$0<\sigma \le \nu (\lambda ):=\frac{(n+2-2q\beta )p^2}{(n+2-2p\alpha )q^2}\lambda$$

$$\sigma =\mu (\lambda ):=\frac{2p\beta }{n+2-2p\alpha }+\frac{n+2}{(n+2-2p\alpha )\lambda }$$

$$({\lambda }_0,{\sigma }_0)=\left(\frac{(n+2)q}{(n+2-2q\beta )p},\frac{(n+2)p}{(n+2-2p\alpha )q}\right).$$

$$A:=\{(\lambda ,\sigma ):0<\sigma \le \nu (\lambda )\text{ and }1/\lambda \le \sigma <\mu (\lambda )\}$$

Full text

[TABLE]

satisfying the initial conditions

[TABLE]

where $\lambda,\sigma,\alpha$, and $\beta$ are positive constants.

Specifically, using the definition of the fractional heat operator $(\partial_{t}-\Delta)^{\alpha}$ given in [24], we obtain, when they exist, optimal pointwise upper bounds on $\mathbb{R}^{n}\times(0,\infty)$ for nonnegative solutions $u$ and $v$ of this initial value problem with particular emphasis on these bounds as $t\to 0^{+}$ and as $t\to\infty$.

2010 Mathematics Subject Classification. 35B09, 35B33, 35B44, 35B45, 35K58, 35R11, 35R45.

Keywords. Blow-up, Pointwise bound, Fractional heat operator, Parabolic system.

1 Introduction

In this paper we study pointwise upper bounds for nonnegative solutions $u(x,t)$ and $v(x,t)$ of the nonlinear system of inequalities

[TABLE]

satisfying the initial conditions

[TABLE]

where $\lambda,\sigma,\alpha$, and $\beta$ are positive constants.

Our results in this paper for the system (1.1), (1.2) are an extension of our results in [24] on pointwise bounds for nonnegative solutions $u(x,t)$ of the scalar initial value problem

[TABLE]

[TABLE]

where $\lambda$ and $\alpha$ are positive constants.

As in [24], we define the fully fractional heat operator

[TABLE]

for

[TABLE]

as the inverse of the operator

[TABLE]

where

[TABLE]

[TABLE]

and

[TABLE]

By (1.6) we mean $X^{p}$ is the set of all measurable functions $f:\mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R}$ such that

[TABLE]

In the definition (1.7) of $J_{\alpha}$,

[TABLE]

is the fractional heat kernel.

When $p$ and $\alpha$ satisfy (1.4), it was shown in [24] that the operator (1.5) has among others the following properties:

1. (P1)

it makes sense because $J_{\alpha}f\in L^{p}_{\text{loc}}(\mathbb{R}^{n}\times\mathbb{R})\text{ for }f\in X^{p}$, 2. (P2)

it is one-to-one and onto, 3. (P3)

if $f\in X^{p}$ and $u=J_{\alpha}f$ then $f=0$ in $\mathbb{R}^{n}\times(-\infty,0)$ if and only if $u=0$ in $\mathbb{R}^{n}\times(-\infty,0)$.

By properties (P1) and (P2) we can indeed define (1.3) as the inverse of (1.5) when $p$ and $\alpha$ satisfy (1.4). Property (P3) will be needed to handle the initial conditions (1.2).

According to our results in Section 2 there are essentially only three possibilities for nonnegative solutions $u\in Y^{p}_{\alpha}$ and $v\in Y^{q}_{\beta}$ of (1.1), (1.2) depending on $n$, $\lambda$, $\sigma$, $\alpha$, $\beta$, $p$, and $q$:

1. (i)

The only solution is $u\equiv v\equiv 0$ in $\mathbb{R}^{n}\times\mathbb{R}$; 2. (ii)

There exist sharp nonzero pointwise bounds for solutions as $t\to 0^{+}$ and as $t\to\infty$; 3. (iii)

There do not exist pointwise bounds for solutions as $t\to 0^{+}$ and as $t\to\infty$.

All possiblities can occur. For the precise statements of possibilities (i), (ii), and (iii) see Theorem 2.1, Theorems 2.2–2.4, and Theorems 2.5 and 2.6, respectively.

The operator (1.3) is a fully fractional heat operator as opposed to time fractional heat operators in which the fractional derivatives are only with respect to $t$, and space fractional heat operators, in which the fractional derivatives are only with respect to $x$.

Some recent results for nonlinear PDEs containing time (resp. space) fractional heat operators can be found in [2, 4, 5, 10, 14, 15, 16, 19, 23, 28, 29] (resp. [1, 3, 7, 8, 9, 11, 12, 13, 17, 20, 25, 26, 27]). Except for [24], we know of no results for nonlinear PDEs containing the fully fractional heat operator $(\partial_{t}-\Delta)^{\alpha}$. However results for linear PDEs containing this operator, including in particular

[TABLE]

where $f$ is a given function, can be found in [6, 18, 21, 22].

2 Statement of Results

In this section we state our results concerning pointwise bounds for nonnegative solutions

[TABLE]

of the nonlinear system of inequalities

[TABLE]

satisfying the initial conditions

[TABLE]

where

[TABLE]

and, as in the definition in Section 1 of the operator (1.3), $p$ and $\alpha$ satisfy

[TABLE]

and $q$ and $\beta$ satisfy

[TABLE]

If $p$ and $\alpha$ satisfy (2.5), $u\in Y^{p}_{\alpha}$, and $(\partial_{t}-\Delta)^{\alpha}u\geq 0$ in $\mathbb{R}^{n}\times\mathbb{R}$ then

[TABLE]

by (1.7) and the nonnegativity of $\Phi_{\alpha}$. Thus the assumption that $u$ (and similarly $v$) is nonnegative can be omitted when studying the problem (2.1)–(2.6).

Moreover, when studying the problem (2.1)–(2.6), we can assume without loss of generality that

[TABLE]

for otherwise switch the symbols for $u,\lambda,\alpha$, and $p$ with the symbols for $v,\sigma,\beta$, and $q$ respectively.

If (2.5) holds then either

[TABLE]

or

[TABLE]

The following Theorems 2.1–2.6 deal with solutions of (2.1)–(2.3) when (2.4)–(2.7) and (2.8) hold; the only exception being that (2.7) and (2.8) are not assumed in Theorems 2.3 and 2.4. Theorem 2.7 deals with solutions of (2.1)–(2.3) in the simpler case when (2.4)–(2.7) and (2.9) hold.

If (2.4), (2.7), and (2.8) hold then

[TABLE]

[TABLE]

and the curves $\sigma=\nu(\lambda)$ and

[TABLE]

intersect at

[TABLE]

See Figure 2.1. Thus assuming (2.4), (2.7), and (2.8) hold, the point $(\lambda,\sigma)$ belongs to one of the following five pairwise disjoint subsets of the $\lambda\sigma$-plane.

[TABLE]

Note that $A,B,C$, and $D$ are two dimensional regions in the $\lambda\sigma$-plane whereas $E$ is the curve separating $A$ and $C$. (See Figure 2.1.)

Theorems 2.1–2.6 deal with solutions of (2.1)–(2.3) when (2.4)–(2.7) and (2.8) hold and $(\lambda,\sigma)$ is in $A,B,C$ or $D$. We have no results when $(\lambda,\sigma)\in E$.

The following theorem deals with solutions of (2.1)–(2.3) when (2.4)–(2.7) and (2.8) hold and $(\lambda,\sigma)\in A$.

Theorem 2.1**.**

Suppose (2.1)–(2.7) and (2.8) hold and

[TABLE]

Then

[TABLE]

The following theorem deals with solutions of (2.1)–(2.3) when (2.4)–(2.7) and (2.8) hold and $(\lambda,\sigma)\in B$.

Theorem 2.2**.**

Suppose (2.1)–(2.7) and (2.8) hold and

[TABLE]

Then for all $T>0$ we have

[TABLE]

and

[TABLE]

where

[TABLE]

[TABLE]

[TABLE]

where $\Gamma$ is the Gamma function.

By the following theorem, the bounds (2.14) and (2.15) in Theorem 2.2 are optimal.

Theorem 2.3**.**

Suppose (2.4)–(2.6) and (2.13) hold,

[TABLE]

where $M_{1}$ and $M_{2}$ are defined in (2.17) and (2.18). Then there exist solutions

[TABLE]

of (2.2), (2.3) such that for $0<t<T$ we have

[TABLE]

where $\gamma_{1}$ and $\gamma_{2}$ are defined in (2.16).

Although the estimates (2.14) and (2.15) are optimal there still remains the question as to whether there is a single solution pair $u,v$ which has the same size as these estimates as $t\to\infty$. By the following theorem there is such a solution pair.

Theorem 2.4**.**

Suppose (2.4)–(2.6) and (2.13) hold. Then there exist $N>0$ and solutions

[TABLE]

of (2.2), (2.3) such that for $|x|^{2}<t$ we have

[TABLE]

where $\gamma_{1}$ and $\gamma_{2}$ are defined in (2.16).

According to the following theorem, if $(\lambda,\sigma)\in C\cup D$ then there exist bounds as $t\to 0^{+}$ for solutions of (2.1)–(2.3) in neither the pointwise (i.e. $L^{\infty}$) sense nor in the $L^{r}$ sense for certain values of $r$. Moreover by Theorem 2.6 the same is true as $t\to\infty$.

Theorem 2.5**.**

Suppose (2.4)–(2.7) and (2.8) hold,

[TABLE]

$r\in(p,\infty]$, and $s\in(s_{0},\infty]$ where

[TABLE]

Then there exist solutions

[TABLE]

of the initial value problem (2.2), (2.3) and a sequence $\{t_{j}\}\subset(0,1)$ such that $\lim_{j\to\infty}t_{j}=0$ and

[TABLE]

[TABLE]

for $j=1,2,\dots$, where

[TABLE]

Since (2.20) implies (2.21) and (2.22) are not true for $r=p$ and $s=q$ respectively, we see that (2.21) is optimal when $(\lambda,\sigma)\in C\cup D$ and (2.22) is optimal when $s_{0}=q$ (i.e. when $(\lambda,\sigma)\in D$).

Theorem 2.6**.**

Suppose (2.4)–(2.7), (2.8), and (2.19) hold. Let

[TABLE]

Then $r_{0}>p$, $s_{0}>q$, and for each

[TABLE]

there exist solutions $u\in Y^{p}_{\alpha}$ and $v\in Y^{q}_{\beta}$ of the initial value problem (2.2), (2.3) and a sequence $\{t_{j}\}\subset(1,\infty)$ such that $\lim_{j\to\infty}t_{j}=\infty$ and $u$ and $v$ satisfy (2.21) and (2.22) for $j=1,2,\dots$, where $R_{j}$ is given by (2.23).

The following theorem deals with solutions of (2.1)–(2.3) when (2.4)–(2.7) and (2.9) hold.

Theorem 2.7**.**

Suppose (2.1)–(2.7) and (2.9) hold. Then the following statements are true.

1. (i)

If $\lambda\sigma\geq 1$ then $u=v=0\text{ a.e. in }\mathbb{R}^{n}\times\mathbb{R}$. 2. (ii)

If $\lambda\sigma<1$ then $u$ and $v$ satisfy (2.14) and (2.15) for all $T>0$.

Clearly Theorem 2.7(i) is optimal and the optimality of Theorem 2.7(ii) follows from Theorems 2.3 and 2.4 because neither (2.8) nor (2.9) is assumed in those theorems.

3 $J_{\alpha}$ version of results

In order to prove our results in Section 2, we will first reformulate them in terms of the inverse $J_{\alpha}$ of the fractional heat operator (1.3) as follows:

If (2.4)–(2.6) hold then by properties (P1)–(P3) in Section 1 of $J_{\alpha}$ and the definition of the fractional heat operator (1.3), $u$ and $v$ satisfy (2.1)–(2.3) if and only if

[TABLE]

satisfy

[TABLE]

[TABLE]

[TABLE]

In problem (3.1)–(3.3), $f$ and $g$ are nonnegative functions in $\mathbb{R}^{n}\times\mathbb{R}$ and thus $J_{\alpha}f$ and $J_{\beta}g$ are well-defined nonnegative extended real valued functions in $\mathbb{R}^{n}\times\mathbb{R}$ without assuming (2.5) and (2.6). Hence in this section we study the problem (3.1)–(3.3) without assumptions (2.5) and (2.6). However our results in this section for the problem (3.1)–(3.3) will only yield corresponding results for the problem (2.1)–(2.3) when (2.5) and (2.6) hold, for otherwise the fractional heat operators in (2.2) are not defined. (For a more detailed discussion of the properties of $J_{\alpha}$ when (2.5) and (2.6) do not hold see [24, Section 4].)

Actually in this section we will consider solutions

[TABLE]

of the following slightly more general version of (3.2)–(3.3):

[TABLE]

[TABLE]

where

[TABLE]

are constants.

As in Section 2, we can assume without loss of generality that

[TABLE]

Under the equivalence of problems (2.1)–(2.3) and (3.1)–(3.3) discussed above, the following Theorems 3.1–3.7 when restricted to the case that (2.5) and (2.6) hold and $K_{1}=K_{2}=1$, clearly imply Theorems 2.1–2.7 in Section 2. We will prove Theorems 3.1–3.7 in Section 5.

If (3.7) holds then either

[TABLE]

or

[TABLE]

Theorem 3.1**.**

Suppose (3.4)–(3.8) and (3.9) hold and

[TABLE]

Then

[TABLE]

Theorem 3.2**.**

Suppose (3.4)–(3.8) and (3.9) hold and

[TABLE]

Then for all $T>0$ we have

[TABLE]

[TABLE]

[TABLE]

and

[TABLE]

where $\gamma_{1},\gamma_{2},M_{1}$, and $M_{2}$ defined in (2.16)–(2.18).

Theorem 3.3**.**

Suppose (3.7) and (3.11) hold,

[TABLE]

where $M_{1}$ and $M_{2}$ are defined in (2.17) and (2.18). Then there exist solutions

[TABLE]

of (3.5), (3.6) such that

[TABLE]

and, for $0<t<T$,

[TABLE]

[TABLE]

[TABLE]

[TABLE]

where $\gamma_{1}$ and $\gamma_{2}$ are defined in (2.16).

Theorem 3.4**.**

Suppose (3.7) and (3.11) hold. Then there exist $N>0$ and solutions $f\in X^{p}$ and $g\in X^{q}$ of (3.5), (3.6) such that for $|x|^{2}<t$ we have

[TABLE]

[TABLE]

where $\gamma_{1}$ and $\gamma_{2}$ are defined in (2.16).

Theorem 3.5**.**

Suppose (3.7), (3.8), and (3.9) hold,

[TABLE]

$r\in(p,\infty]$, and $s\in(s_{0},\infty]$, where

[TABLE]

Then there exist solutions

[TABLE]

of the initial value problem (3.5), (3.6) and a sequence $\{t_{j}\}\subset(0,1)$ such that $\lim_{j\to\infty}t_{j}=0$ and

[TABLE]

where

[TABLE]

Theorem 3.6**.**

Suppose (3.7), (3.8), (3.9) and (3.25) hold. Let

[TABLE]

Then $r_{0}>p$, $s_{0}>q$, and for each

[TABLE]

there exist solutions

[TABLE]

of the initial value problem (3.5), (3.6) and a sequence $\{t_{j}\}\subset(1,\infty)$ such that $\lim_{j\to\infty}t_{j}=\infty$ and $f$ and $g$ satisfy (3.27) where $R_{j}$ is given by (3.28).

Theorem 3.7**.**

Suppose (3.4)–(3.8) and (3.10) hold. Then the following statements are true.

1. (i)

If $\lambda\sigma\geq 1$ then $f=g=0\text{ a.e. in }\mathbb{R}^{n}\times\mathbb{R}$. 2. (ii)

If $\lambda\sigma<1$ then $f$ and $g$ satisfy (3.12)–(3.15) for all $T>0$.

4 Preliminarys

In the section we provide some remarks and lemmas needed for the proofs of our results in Section 3 dealing with solutions of the $J_{\alpha}$ problem (3.4)–(3.7).

Remark 4.1**.**

If (3.7)2 holds and $\lambda\sigma<1$ then the functions $F,G:\mathbb{R}^{n}\times\mathbb{R}\to[0,\infty)$ defined in $\mathbb{R}^{n}\times(-\infty,0]$ by $F=G=0$ and defined for $(x,t)\in\mathbb{R}^{n}\times(0,\infty)$ by

[TABLE]

where $\gamma_{1},\gamma_{2},M_{1}$, and $M_{2}$ are defined in (2.16)–(2.18), satisfy

[TABLE]

which can be verified using the formula

[TABLE]

Even though $F,G\notin X^{p}$ for all $p\geq 1$, these functions will be useful in our analysis of solutions of (3.5), (3.6) which are in $X^{p}$ for some $p\geq 1$.

Remark 4.2**.**

It will be convenient to scale (3.5) as follows. Suppose (3.7)2 holds, $\lambda\sigma\neq 1$, $T>0$, and $f,g,\bar{f},\bar{g}:\mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R}$ are nonnegative measurable functions such that $f=g=\bar{f}=\bar{g}=0$ in $\mathbb{R}^{n}\times(-\infty,0)$ and

[TABLE]

where $\gamma_{1}$ and $\gamma_{2}$ are defined in (2.16) and $x=T^{1/2}\bar{x}$ and $t=T\bar{t}$. Then $f$ and $g$ satisfy (3.5) if and only if $\bar{f}$ and $\bar{g}$ satisfy

[TABLE]

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

[TABLE]

Lemma 4.1**.**

Suppose (3.4)–(3.7) and (3.10) hold. Then

[TABLE]

Proof.

Let $T>0$ be fixed. To prove (4.3) it suffices by (3.6) to prove

[TABLE]

Choose

[TABLE]

Define $\widehat{\alpha}\in\mathbb{R}$ by

[TABLE]

by (3.10). Then $0<\widehat{\alpha}<\alpha$ and

[TABLE]

by (3.7)1. Hence by (3.6), (3.4)1, and Lemma A.2 we have

[TABLE]

Consequently by (3.5)2 we have

[TABLE]

By (4.4) there exists $\widehat{\beta}\in(0,\beta]$ such that

[TABLE]

Then by (3.5)1, (3.6), (4.5), Lemma A.2 we have

[TABLE]

Thus by (3.5)2, (3.6), and Lemma A.1 we have

[TABLE]

∎

Lemma 4.2**.**

Suppose (3.4)–(3.7) and (3.9) hold,

[TABLE]

[TABLE]

Then either

[TABLE]

or there exists a constant

[TABLE]

such that $f\in X^{p_{2}}$ for some $p_{2}\in(p_{1},\infty)$ satisfying

[TABLE]

Proof.

If $2p_{1}\alpha\geq n+2$ then (4.8) follows from Lemma 4.1 with $p=p_{1}$. Hence we can assume

[TABLE]

By (3.7), (3.9), and (4.6) there exists

[TABLE]

such that

[TABLE]

and

[TABLE]

By (4.11) and (4.12) we have

[TABLE]

By (4.11), (4.12), (3.7), and (4.7) there exists $p_{3}\in(p_{1},\infty)$ such that

[TABLE]

Hence by (3.6), (4.7), and Lemma A.2 we have $J_{\alpha}f\in X^{p_{3}}$ and thus from (3.5) and (3.4) we find that

[TABLE]

by (3.7). We can assume

[TABLE]

for otherwise by Lemma 4.1 with $q=p_{4}$ and the roles of $(f,p,\lambda,\alpha)$ and $(g,q,\sigma,\beta)$ interchanged, we have (4.8) holds.

It follows from (4.17), (4.12), and (4.16) that there exists $p_{5}\in(p_{4},\infty)$ such that

[TABLE]

Thus by (3.6), (4.16) and Lemma A.2 we have $J_{\beta}g\in X^{p_{5}}$ and consequently by (3.5)

[TABLE]

Moreover, it follows from (4.19), (4.18), (4.16), (4.15), (4.14), and (4.7)1 that

[TABLE]

by (4.13). ∎

Lemma 4.3**.**

Suppose (3.4)–(3.7) and (3.9) hold and $\sigma<\mu(\lambda)$. Then

[TABLE]

Proof.

Starting with the assumption that $f$ satisfies (3.4)1 and iterating Lemma 4.2 a finite number of times ($m$ times is enough if $m>1/(pC_{0})$) we find that $f\in X^{\infty}$ and hence (4.20) follows from (3.5), (3.6), and Lemma A.1. ∎

Lemma 4.4**.**

Suppose $f,g:\mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R}$ are nonnegative measurable functions satisfying (3.5) and (3.6) where

[TABLE]

and for some $h\in\{f,g\}$ we have

[TABLE]

Then $\lambda\sigma<1$ and $f$ and $g$ satisfy (3.12)–(3.15) for all $T>0$.

Proof.

Suppose $h=f$. The proof when $h=g$ is similar and will be omitted. By (3.5) we have

[TABLE]

and by (3.6) and (4.21) there exists $a\geq 0$ such that

[TABLE]

and

[TABLE]

Thus

[TABLE]

Let $T>a$ be momentarily fixed. Then for $(x,t)\in\mathbb{R}^{n}\times(a,T]$ we find from (4.23) and (4.24) that

[TABLE]

Hence for $(x,t)\in\mathbb{R}^{n}\times(a,T]$ we obtain from (4.25) and (4.2)that

[TABLE]

where

[TABLE]

Thus by (4.22) we see that

[TABLE]

which by (4.24) implies

[TABLE]

Thus $\lambda\sigma<1$ for otherwise sending $T$ to $a$ in (4.26) gives a contradiction. Hence from (4.26) and (2.16)1 we get

[TABLE]

which together with (4.23) and the nonnegativity of $a$ implies

[TABLE]

Suppose for some $\delta>0$ we have

[TABLE]

Then by (4.23) and (4.2) we have for $(x,t)\in\mathbb{R}^{n}\times(0,\infty)$ that

[TABLE]

and hence by (4.25), (4.2), and (2.17) we find for $(x,t)\in\mathbb{R}^{n}\times(0,\infty)$ that

[TABLE]

because by (2.16) we have

[TABLE]

Thus by (4.22)

[TABLE]

Next defining a sequence $\{\delta_{j}\}\subset(0,\infty)$ by

[TABLE]

and using $0<\lambda\sigma<1$ we see that $\delta_{j}\to(K_{1}K_{2}^{\lambda})^{\frac{1}{1-\lambda\sigma}}M^{\frac{\lambda\sigma}{1-\lambda\sigma}}_{1}$ as $j\to\infty$. It therefore follows from (4.27), (4.28), and (4.31) that $f$ satisfies (3.12) for $T>0$. Thus (4.29) holds with $\delta=(K_{1}K_{2}^{\lambda})^{\frac{1}{1-\lambda\sigma}}M^{\frac{\lambda\sigma}{1-\lambda\sigma}}_{1}$ and so from (3.5)2 we find that

[TABLE]

by (4.30), (2.17), and (2.18). That is $g$ satisfies (3.13) for $T>0$. Finally, (3.14) and (3.15) follow from (3.12), (3.13), and (4.2). ∎

5 Proofs of results for $J_{\alpha}$ problem.

In this section we prove our results stated in Section 3 concerning pointwise bounds for nonnegative solutions $f$ and $g$ of (3.4)–(3.7). As explained in Section 3, these results immediately imply Theorems 2.1–2.7 in Section 2.

Proof of Theorem 3.7 (resp. Theorems

3.1 and 3.2).

It follows from Lemma 4.1 (resp. Lemma 4.3) that $f,g\in X^{\infty}$. Hence Theorem 3.7 (resp. Theorems 3.1 and 3.2) follow(s) from Lemma 4.4. ∎

Proof of Theorem 3.3.

Let

[TABLE]

and $a_{1}=(m+1)/2$. By (3.11) and (3.16) we have $m\in(0,1)$ and thus

[TABLE]

It therefore follows from (3.11) that $a^{1/\lambda}_{1}<a^{\sigma}_{1}$ and hence there exists $a_{2}$ such that

[TABLE]

which together with (5.1) and (5.2) gives

[TABLE]

[TABLE]

By Remark 4.2, we can assume $K_{1}=K_{2}=T=1$. For $(x,t)\in\mathbb{R}^{n}\times\mathbb{R}$ and $\delta\in(0,1)$ let

[TABLE]

where $F$ and $G$ are as in Remark 4.1 and $\psi_{\delta}\in C^{\infty}(\mathbb{R}\to[0,1])$ satisfies

[TABLE]

Then for $1\leq t\leq 1+\delta$

[TABLE]

and similarly

[TABLE]

Thus by (4.1) we have for $1\leq t\leq 1+\delta$ that

[TABLE]

and similarly for $1\leq t\leq 1+\delta$ that

[TABLE]

where $C=C(\lambda,\sigma,\alpha,\beta)>0$. Hence choosing $\delta\in(0,1)$ sufficiently small and using (4.1) and (5.3) we find for $1\leq t\leq 1+\delta$ that

[TABLE]

and

[TABLE]

which by (5.5) and (4.1) holds for all other $t$ as well.

Next let $\varphi(x)=e^{-\psi(x)}$ where $\psi(x)=\sqrt{1+|x|^{2}}-1$. Then for $\varepsilon\in(0,1)$, $\gamma>1$, and $|\xi-x|<\gamma\sqrt{2}$ we have

[TABLE]

Thus defining $f_{\varepsilon},g_{\varepsilon}:\mathbb{R}^{n}\times\mathbb{R}\to[0,\infty)$ by

[TABLE]

we find for $|\xi-x|<\gamma\sqrt{2}$ and $\tau\in\mathbb{R}$ that

[TABLE]

Hence for $(x,t)\in\mathbb{R}^{n}\times(0,2)$ we have

[TABLE]

and

[TABLE]

But for $x,\xi\in\mathbb{R}^{n}$ and $0<\tau<t<2$ we find making the change of variables $z=\frac{x-\xi}{\sqrt{4(t-\tau)}}$ that

[TABLE]

as $\gamma\to\infty$. Thus by (5.8), (5.9), (5.6) and (5.7) we have for $(x,t)\in\mathbb{R}^{n}\times(0,1+\delta)$ that

[TABLE]

and

[TABLE]

by (3.11).

So first choosing $\gamma$ so large that

[TABLE]

(we can do this by (5.3) and (5)) and then choosing $\varepsilon>0$ so small that (5) and (5) are both greater than one we see that $f:=f_{\varepsilon}$ and $g=g_{\varepsilon}$ satisfy (3.5) in $\mathbb{R}^{n}\times(0,1+\delta)$. Hence, since $F_{\delta}$ and $G_{\delta}$, and thus $f(x,t)$ and $g(x,t)$, are identically zero in $\mathbb{R}^{n}\times((-\infty,0]\cup[1+\delta,\infty))$ we have that $f$ and $g$ satisfy (3.5), (3.6).

From the exponential decay of $\varphi(x)$ as $|x|\to\infty$, we find that $f$ and $g$ satisfy (3.17). Also since $f$ and $g$ are uniformly continuous and bounded on $\mathbb{R}^{n}\times\mathbb{R}$ and

[TABLE]

we easily check that (3.18) holds.

Finally, from (5.4) we see for $0<t<1$ that

[TABLE]

and

[TABLE]

and consequently we obtain (3.19) and (3.20). Hence (3.21) and (3.22) follow from (3.5), (3.6). ∎

Proof of Theorem 3.4.

By Remark 4.2 with $T=1$ we can assume $K_{1}=K_{2}=1$. Define $\bar{f},\bar{g}:\mathbb{R}^{n}\times\mathbb{R}\to[0,\infty)$ by

[TABLE]

where $F$ and $G$ are defined in Remark 4.1 and $\Omega=\{|x|^{2}<t\}$. Then using Lemma A.3 and the fact that $\alpha+\gamma_{1}=\gamma_{2}/\sigma$ we obtain for $|x|^{2}<t$ that

[TABLE]

which also holds in $\mathbb{R}^{n}\times\mathbb{R}\backslash\Omega$ because $\bar{g}=0$ there. Similarly

[TABLE]

Thus letting

[TABLE]

for appropriately chosen positive constants $L_{1}$ and $L_{2}$, the function $f$ and $g$ will satisfy (3.4)–(3.6).

It follows from (5.13), (5.14) and the definition of $F$ and $G$ in Remark 4.1 that there exists $N>0$ such that (3.23) holds in $\Omega$. Thus, since $f$ and $g$ solve (3.5) we obtain (3.24), provided we decrease $N$ if necessary. ∎

Remark 5.1**.**

Suppose (3.7), (3.8), (3.9), and (3.25) hold. We will need for the proof of Theorems 3.5 and 3.6 some observations concerning the graphs of the straight lines in the $\xi\eta$-plane given by

[TABLE]

These lines intersect the vertical line $\xi=\frac{n+2}{2p}$ at

[TABLE]

respectively, where

[TABLE]

by (2.12). Thus

[TABLE]

Moreover, it follows from (5.16) and (3.25) that

[TABLE]

and it follows from (3.8) and (3.25) (see Figure 2.1) that $\lambda>\lambda_{0}$ where $\lambda_{0}$ is defined in (2.12). Thus by (5.16)

[TABLE]

The lines (5.15) are graphed in Figures 5.1a, 5.1b, and 5.1c when $\sigma<\sigma_{0},\sigma=\sigma_{0}$, and $\sigma>\sigma_{0}$ respectively.

Proof of Theorem 3.5.

Since $|R_{j}|<\infty$ for $j=1,2,\dots$, to prove Theorem 3.5 it suffices to show for each $\varepsilon>0$ there exist

[TABLE]

such that the conclusion of Theorem 3.5 holds.

We will use the notation and observations in Remark 5.1. Let $P_{0}=(\xi_{0},\eta_{0})$ where

[TABLE]

Then

[TABLE]

The point $P_{0}$ is graphed in Figure 5.1. It follows from Figure 5.1 that there exist points $P_{1}(\xi_{1},\eta_{1})$ in the open shaded region arbitrarily close to $P_{0}$. More precisely, fixing $\varepsilon>0$, there exist $\xi_{1}\in(0,\xi_{0})$ and $\eta_{1}\in(0,\eta_{0})$ such that

[TABLE]

and

[TABLE]

Thus defining $r$ and $s$ by

[TABLE]

we have $r$ and $s$ satisfy (5.19).

Define $f_{0},g_{0}:\mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R}$ by

[TABLE]

where

[TABLE]

Then by (5.21) and Lemma A.5 we have

[TABLE]

and for $(x,t)\in\Omega_{0}$

[TABLE]

where $C=(n,\lambda,\sigma,\alpha,\beta,p,q,r,s)>0$.

Let $\{T_{j}\}\subset(0,1/2)$ be a sequence such that

[TABLE]

and define

[TABLE]

Then

[TABLE]

Defining $f_{j},g_{j}:\mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R}$ by

[TABLE]

we obtain from (5.21), (5.22), (5.27), and Lemma A.6 that

[TABLE]

[TABLE]

and for $(x,t)\in\Omega^{+}_{j}:=\{(x,t)\in\Omega_{j}:\frac{3T_{j}}{4}<t<T_{j}\}$ that

[TABLE]

It follows from (5.23) and (5.25) that for $(x,t)\in\Omega_{0}$ we have

[TABLE]

and

[TABLE]

Thus by (5.20) and (5.27) we find that

[TABLE]

and

[TABLE]

by taking a subsequence.

Using (5.28), (5.31), (5.20) and taking a subsequence we obtain

[TABLE]

and similarly

[TABLE]

It follows from (5.23), (5.28),(5.27), and (5.26) that

[TABLE]

and

[TABLE]

and letting $\Omega^{-}_{j}=\Omega_{j}\backslash\Omega^{+}_{j}$, using (5.25), (5.28), (5.27), (5.26) and (5.20) and taking a subsequence we obtain

[TABLE]

and similarly

[TABLE]

Taking an appropriate subsequence of $(f_{j},g_{j})$ and letting

[TABLE]

we see from (5.24) and (5.29) that (3.26) holds. In $\Omega^{+}_{j}$ we have by (5.33) and (5) that

[TABLE]

and similarly by (5.33) and (5.35) that $g\leq C(\sigma)(J_{\alpha}f)^{\sigma}$.

In $\Omega^{-}_{j}$ we have by (5.36) and (5) that

[TABLE]

and similarly by (5.37) and (5.39) that $g\leq(J_{\alpha}f)^{\sigma}$. In $\Omega_{0}\backslash\bigcup^{\infty}_{j=1}\Omega_{j}$ we have by (5.32) that

[TABLE]

and

[TABLE]

In $\mathbb{R}^{n}\times\mathbb{R}\backslash\Omega_{0}$, $f=0\leq(J_{\beta}g)^{\lambda}$ and $g=0\leq(J_{\alpha}f)^{\sigma}$. Thus, after scaling $f$ and $g$ we see that $f$ and $g$ are solutions of (3.5) and (3.6). Also (3.27) holds by (5.30). ∎

Proof of Theorem 3.6.

We will use the notation and observations in Remark 5.1. Let $P_{4}=(\xi_{4},\eta_{4})$ be the point where the lines (5.15) intersect. It follows from (5.17) and (5.18) (see Figure 5.1) that

[TABLE]

Thus, since solving the system (5.15) yields

[TABLE]

we see that $r_{0}>p$ and $s_{0}>q$.

Define $f_{0},g_{0}:\mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R}$ by

[TABLE]

where $\Omega_{0}$ is as in Lemma A.5. Then by (5.40), Lemma A.5 and the fact that $P_{4}$ satisfies (5.15) we have

[TABLE]

and

[TABLE]

where in this proof $C$ is a positive constant depending on $n,\lambda,\sigma,\alpha,\beta,p,q$, whose values may change from line to line.

Let $\{T_{j}\},\{t_{j}\}\subset(2,\infty)$ satisfy $T_{j+1}>4T_{j}$ and $T_{j}=2t_{j}$ and define $f_{j},g_{j}:\mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R}$ by

[TABLE]

where

[TABLE]

Then

[TABLE]

[TABLE]

and by (5.44), (5.40), Lemma A.6, and the fact that $P_{4}$ satisfies (5.15) we have

[TABLE]

and

[TABLE]

where

[TABLE]

It follows therefore from (5.43) that

[TABLE]

and similarly

[TABLE]

In $\Omega^{-}_{j}=\Omega_{j}\backslash\Omega^{+}_{j}$ we have

[TABLE]

and similarly

[TABLE]

Thus we obtain from (5.43) that

[TABLE]

and similarly that

[TABLE]

Let

[TABLE]

Then clearly $f$ and $g$ satisfy (3.6) and by (5.42), (5.47), and (5.46) we see that $f$ and $g$ satisfy (3.30).

In $\Omega_{j}$ we have by (5.45)2, (5.48), (5.49), (5.50), and (5.51) that

[TABLE]

and in $(\mathbb{R}^{n}\times\mathbb{R})\backslash\cup^{\infty}_{j=1}\Omega_{j}$ we have by (5.43) that

[TABLE]

and

[TABLE]

Thus after scaling $f$ and $g$, we find that $f$ and $g$ satisfy (3.5).

From (5.41), (5.44), (5.45)1 and Lemma A.6 we find

[TABLE]

and

[TABLE]

Thus, since $|R_{j}|<\infty$, we have (3.27) holds for all $r$ and $s$ satisfying (3.29). ∎

Appendix A Auxiliary lemmas

In this appendix we provide some lemmas needed for the proofs of our results in Section 3 dealing with solutions of the $J_{\alpha}$ problem (3.4)–(3.7). See [24, Section 7] for the proofs of these lemmas.

Let $\Omega=\mathbb{R}^{n}\times(a,b)$ where $n\geq 1$ and $a<b$. The following two lemmas give estimates for the convolution

[TABLE]

where $\alpha>0$ and $\Phi_{\alpha}$ is defined in (1.9).

Remark A.1**.**

Note that if $f:\mathbb{R}^{n}\times\mathbb{R}\to\mathbb{R}$ is a nonnegative measurable function such that $\|f\|_{L^{\infty}(\mathbb{R}^{n}\times\mathbb{R}_{a})}=0$ then

[TABLE]

Lemma A.1**.**

For $\alpha>0,\,\Omega=\mathbb{R}^{n}\times(a,b)$ and $f\in L^{\infty}(\Omega)$ we have

[TABLE]

Lemma A.2**.**

Let $p,q\in[1,\infty]$, $\alpha$, and $\delta$ satisfy

[TABLE]

Then $V_{\alpha,\Omega}$ maps $L^{p}(\Omega)$ continuously into $L^{q}(\Omega)$ and for $f\in L^{p}(\Omega)$ we have

[TABLE]

where

[TABLE]

Lemma A.3**.**

Suppose $x\in\mathbb{R}^{n}$ and $t,\tau\in(0,\infty)$ satisfy

[TABLE]

Then

[TABLE]

where $\Phi_{\alpha}$ is defined by (1.9).

Lemma A.4**.**

For $\tau<t\leq T$ and $|x|\leq\sqrt{T-t}$ we have

[TABLE]

where $C=C(n)$ is a positive constant.

Lemma A.5**.**

Suppose $\alpha>0$, $\gamma>0$, $p\geq 1$, and

[TABLE]

Then $f_{0}\in X^{p}$ and

[TABLE]

where $C_{1}$ and $C_{2}$ are positive constants depending only on $n,\alpha,\gamma$, and $p$.

Lemma A.6**.**

Suppose $\alpha>0$, $\gamma\in\mathbb{R}$, $0\leq t_{0}<T,\,p\in[1,\infty)$, and

[TABLE]

where

[TABLE]

Then

[TABLE]

for $(x,t)\in\Omega^{+}:=\{(x,t)\in\Omega:\frac{T+t_{0}}{2}<t<T\}$ where $C=C(n,\alpha,\gamma,p)>0$. Moreover,

[TABLE]

and in this case

[TABLE]