Discontinuous critical Fujita exponents for the heat equation with combined nonlinearities

TL;DR

This paper investigates the blow-up and global existence of solutions to a nonlinear heat equation with combined nonlinearities, revealing a discontinuous critical Fujita exponent influenced by the gradient term.

Contribution

It establishes a novel discontinuity in the critical Fujita exponent for positive solutions and extends results to sign-changing solutions, highlighting the gradient term's impact.

Findings

Finite time blow-up for certain exponents p and q.

Existence of global solutions for small initial data.

Discontinuity of the critical Fujita exponent as q varies.

Abstract

We consider the nonlinear heat equation $u_t-\Delta u =|u|^p+b |\nabla u|^q$ in $(0,\infty)\times \R^n$, where $n\geq 1$, $p>1$, $q\geq 1$ and $b>0$. First, we focus our attention on positive solutions and obtain an optimal Fujita-type result: any positive solution blows up in finite time if $p\leq 1+\frac{2}{n}$ or $q\leq 1+\frac{1}{n+1}$, while global classical positive solutions exist for suitably small initial data when $p>1+\frac{2}{n}$ and $q> 1+\frac{1}{n+1}$. Although finite time blow-up cannot be produced by the gradient term alone and should be considered as an effect of the source term $|u|^p$, this result shows that the gradient term induces an interesting phenomenon of discontinuity of the critical Fujita exponent, jumping from $p=1+\frac{2}{n}$ to $p=\infty$ as $q$ reaches the value $1+\frac{1}{n+1}$ from above. Next, we investigate the case of sign-changing solutions and show that if $p\le 1+\frac{2}{n}$ or $0<(q-1)(np-1)\le 1$, then the solution blows up in finite time for any nontrivial initial data with nonnegative mean. Finally, a Fujita-type result, with a different critical exponent, is % also obtained for sign-changing solutions to the inhomogeneous version of this problem.

Equations

$$\displaystyle\left\{\begin{array}[]{lllll}u_{t}-\Delta u&=&|u|^{p}+b|\nabla u|^{q}&\mbox{in}&(0,\infty)\times\mathbb{R}^{n},\vspace{1mm}\\ u(0,x)&=&u_{0}(x)&\mbox{in}&\mathbb{R}^{n},\end{array}\right.$$

$$\displaystyle\left\{\begin{array}[]{lllll}u_{t}-\Delta u&=&|u|^{p}+b|\nabla u|^{q}+h(x)&\mbox{in}&(0,\infty)\times\mathbb{R}^{n},\vspace{1mm}\\ u(0,x)&=&u_{0}(x)&\mbox{in}&\mathbb{R}^{n}.\end{array}\right.$$

$$BC^{1}(\mathbb{R}^{n})=\bigl{\{}\phi\in C^{1}(\mathbb{R}^{n}):\,\phi,\nabla\phi\in L^{\infty}(\mathbb{R}^{n})\bigr{\}}.$$

$$T<\infty ⟹\underset{t\to T^{-}}{\lim }\Vert u(t){\Vert }_{C^1}=\infty .$$

$$\displaystyle\left\{\begin{array}[]{lllll}u_{t}-\Delta u&=&|u|^{p}&\mbox{in}&(0,\infty)\times\mathbb{R}^{n},\vspace{1mm}\\ u(0,x)&=&u_{0}(x)&\mbox{in}&\mathbb{R}^{n}.\end{array}\right.$$

$$p_F=1+\frac{2}{n}$$

$$\displaystyle\left\{\begin{array}[]{lllll}u_{t}-\Delta u&=&|\nabla u|^{q}&\mbox{in}&(0,\infty)\times\mathbb{R}^{n},\vspace{1mm}\\ u(0,x)&=&u_{0}(x)&\mbox{in}&\mathbb{R}^{n}.\end{array}\right.$$

$$\Vert u(t){\Vert }_{\infty }\le \Vert u_0{\Vert }_{\infty }\text{mbox}and\Vert \nabla u(t){\Vert }_{\infty }\le \Vert \nabla u_0{\Vert }_{\infty },t\ge 0.$$

$$\displaystyle\left\{\begin{array}[]{lllll}u_{t}-\Delta u&=&|u|^{p-1}u-b|\nabla u|^{q}&\mbox{in}&(0,\infty)\times\mathbb{R}^{n},\vspace{1mm}\\ u(0,x)&=&u_{0}(x)&\mbox{in}&\mathbb{R}^{n},\end{array}\right.$$

$$u_{tt}-\Delta u=|u{|}^p+|u_t{|}^q,t>0,x\in {\mathbb{R}}^n,$$

$$p\le p_{F}orq\le 1+\frac{1}{n+1},$$

$$p>p_{F}andq>1+\frac{1}{n+1}.$$

$$0\le u_0(x)\le \epsilon e^{-\frac{|x{|}^2}{4}}\text{mbox}and{u}_0\not\equiv 0,$$

$$\displaystyle\left\{\begin{array}[]{lllll}u_{t}-\Delta u&=&|u|^{p-1}u+a\cdot\nabla(u^{q})&\mbox{in}&(0,\infty)\times\mathbb{R}^{n},\vspace{1mm}\\ u(0,x)&=&u_{0}(x)&\mbox{in}&\mathbb{R}^{n},\end{array}\right.$$

$$p_1(n,q):=\{\begin{array}{ll}1+\frac{2}{n},& ifq\ge \frac{n+1}{n},\\ 1+\frac{2q}{n+1},& if\; 1<q<\frac{n+1}{n},\\ 1+\frac{2}{n},& ifq=1.\end{array}$$

$$u(t,x)\le M,(t,x)\in (0,\min (T,\tau ))\times {\mathbb{R}}^n,$$

$$|\nabla u(t,x)|\le N,(t,x)\in (0,\min (T,\tau ))\times {\mathbb{R}}^n.$$

$$\limsup_{t\to T^{-}}\,\Bigl{(}\sup_{x\in\mathbb{R}^{n}}u(t,x)\Bigr{)}=\infty.$$

$$\underset{t\in (0,T),x\in {\mathbb{R}}^n}{\inf }u(t,x)\ge m:=\underset{x\in {\mathbb{R}}^n}{\inf }{u}_0(x)>-\infty ,$$

$$|u(t,x)|\le \max (M,|m|),(t,x)\in (0,\min (T,\tau ))\times {\mathbb{R}}^n.$$

$$z_i(t,x):=\sigma {\partial }_{x_i}u(t,x),(t,x)\in (0,\min (T,\tau ))\times {\mathbb{R}}^n,i\in \{1,\cdots ,n\}.$$

$$(z_i{)}_t-\Delta z_i=p|u{|}^{p-2}u{z}_i+bq|\nabla u{|}^{q-2}\nabla u\cdot \nabla z_i+\sigma {\partial }_{x_i}h(x),(t,x)\in (0,\min (T,\tau ))\times {\mathbb{R}}^n,$$

$$(z_i{)}_t-\Delta z_i\le K|z_i|+q|b||\nabla u{|}^{q-1}|\nabla z_i|+|{\partial }_{x_i}h|,(t,x)\in (0,\min (T,\tau ))\times {\mathbb{R}}^n,$$

$$\overline{z}_{i}(t,x):=\bigl{(}\|z_{i}(0,\cdot)\|_{\infty}+K^{-1}\|\partial_{x_{i}}h\|_{\infty}\bigr{)}e^{Kt},\quad(t,x)\in(0,\min(T,\tau))\times\mathbb{R}^{n}$$

$$\pm\partial_{x_{i}}u(t,x)\leq\bigl{(}\|z_{i}(0,\cdot)\|_{\infty}+K^{-1}\|\partial_{x_{i}}h\|_{\infty}\bigr{)}e^{K\tau},\quad(t,x)\in(0,\min(T,\tau))\times\mathbb{R}^{n},\,\,i\in\{1,\cdots,n\},$$

$$\displaystyle\left\{\begin{array}[]{lllll}v_{t}-\Delta v&=&v^{p}&\mbox{in}&(0,\infty)\times\mathbb{R}^{n},\\ v(0,x)&=&u_{0}(x)&\mbox{in}&\mathbb{R}^{n}\end{array}\right.$$

$$\displaystyle\left\{\begin{array}[]{lllll}w_{t}-\Delta w&=&b|\nabla w|^{q}&\mbox{in}&(0,\infty)\times\mathbb{R}^{n},\\ w(0,x)&=&u_{0}(x)&\mbox{in}&\mathbb{R}^{n}.\end{array}\right.$$

$$\underset{t\to \infty }{\lim }w(t,x)=\ell ,uniformly\; on\; compact\; subsets\; of{\mathbb{R}}^n.$$

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

$$u(t_0,x)\ge w(t_0,x)\ge \frac{\ell }{2},for\; allx\in B_R,$$

Full text

1. Introduction

Consider the problem

[TABLE]

and its inhomogeneous version

[TABLE]

Here $p>1$, $q\geq 1$, $b>0$ and $0\leq h\in BC^{1}(\mathbb{R}^{n})$, where

[TABLE]

Recall (see e.g. [30, Section 35]) that problem (1.6) ((1.3) is a special case) is locally well posed 111Local well-posedness remains true under weaker regularity assumptions on the initial data; however we shall not discuss this here, since our focus is on the large time behavior of solutions. for $u_{0}\in BC^{1}(\mathbb{R}^{n})$. Namely, there exists $\sigma>0$ such that problem (1.6) admits a unique local classical solution $u\in C^{1,2}((0,\sigma)\times\mathbb{R}^{n})$ with $u,\nabla u\in BC([0,\sigma)\times\mathbb{R}^{n})$, where $BC([0,\sigma)\times\mathbb{R}^{n})$ denotes the space of bounded and continuous functions in $[0,\sigma)\times\mathbb{R}^{n}$. Denoting by $T=T(u_{0})\in(0,\infty]$ the existence time of the (unique) maximal classical solution, we have blow-up alternative in $C^{1}$ norm, i.e.

[TABLE]

Moreover, we have $u>0$ in $(0,T)\times\mathbb{R}^{n}$ whenever $u_{0}\geq 0$ and $u_{0}\not\equiv 0$. Actually, it can be shown that the $L^{\infty}$ norm cannot stay bounded if $T<\infty$ (see Proposition 2.1). The first aim of this paper is to give optimal Fujita-type results for problems (1.3) and (1.6). We mention below some motivations for studying such problems.

In the case $b=0$, problem (1.3) reduces to the semilinear heat equation

[TABLE]

In [14], Fujita proved the following results for problem (1.10):

• $\bullet$

If $1<p<1+\frac{2}{n}$, then problem (1.10) admits no global positive solutions;

• $\bullet$

If $p>1+\frac{2}{n}$, then for positive initial values bounded by a sufficiently small Gaussian, problem (1.10) admits positive global solutions.

Later, it was shown that $1+\frac{2}{n}$ belongs to the blow-up case (see [3, 17, 33, 20, 34]). The exponent

[TABLE]

is said to be critical in the sense of Fujita. Note that in [34], it was also shown that if $p>p_{F}$ and $u_{0}\in L^{\frac{n(p-1)}{2}}(\mathbb{R}^{n})$ is sufficiently small, then we have global positive solutions.

Consider now the problem

[TABLE]

Problem (1.14) is the well-known viscous Hamilton-Jacobi equation. Such type of equations arise in stochastic control theory (see [26]) and have also been used to describe a model for growing random interfaces (see e.g. [19, 21]). On the other hand, it is known that if $u_{0}\in BC^{1}(\mathbb{R}^{n})$, problem (1.14) admits a unique local in time solution, classical for $t>0$. Moreover, $u$ exists globally for all $t\geq 0$, and satisfies

[TABLE]

For more details, see e.g. [2, 5, 6, 7, 9, 16]. The large time behavior of positive solutions to problem (1.14) was investigated in [7, 23, 8, 15]. In particular, under the assumption $u_{0}\geq 0$, $u_{0}\not\equiv 0$ and $u_{0}\in BC^{1}(\mathbb{R}^{n})\cap L^{1}(\mathbb{R}^{n})$, denoting $I_{\infty}=\lim_{t\to\infty}\|u(t)\|_{1}$, it was shown in [23] that

• $\bullet$

If $1\leq q\leq\frac{n+2}{n+1}$, then $I_{\infty}=\infty$, for all $u_{0}$;

• $\bullet$

If $\frac{n+2}{n+1}<q<2$, then both $I_{\infty}=\infty$ and $I_{\infty}<\infty$ occur;

• $\bullet$

If $q\geq 2$, then $I_{\infty}<\infty$, for all $u_{0}$.

Moreover, a classification of the pointwise asymptotic behaviors of positive solutions of (1.14) as $t\to\infty$ was obtained in [8, 15].

Next, consider the problem

[TABLE]

where $p>1$, $q\geq 1$ and $b>0$. It is known that problem (1.17) admits a unique, maximal in time, classical solution $u\geq 0$, for all $u_{0}\in BC^{1}(\mathbb{R}^{n})$ with $u_{0}\geq 0$. Problem (1.17) (also in bounded domains with Dirichlet boundary conditions) was first proposed in [11] in order to investigate the possible effect of a damping gradient term on global existence or nonexistence. On the other hand, a model in population dynamics was proposed in [31], where this problem describes the evolution of the population density of a biological species, under the effect of certain natural mechanisms. For a detailled review of this problem, we refer to the survey paper [32]. For any $p>p_{F}$ and $b>0$ (and any $q$), it follows from a comparison argument with the case $b=0$ that there always exist positive global solutions to problem (1.17). Moreover, if $1<q<\frac{2p}{p+1}$ or if $q=\frac{2p}{p+1}$ and $b$ is large, then both blowing-up and stationary positive solutions do exist for any $p>1$. Also some global positive solutions exist when $q\geq p>1$. Therefore, no Fujita-type phenomenon occurs in these cases. As an answer to an open problem proposed in [32], it was shown in [29], by a test function approach, that if $q=\frac{2p}{p+1}$, $1<p<p_{F}$ and $b$ is small enough, then problem (1.17) admits no positive global solutions. Hence, $p_{F}$ is critical in this case. As for the remaining range $\frac{2p}{p+1}<q<p$ and $p>p_{F}$, it seems to be unknown whether (1.17) admits any global positive solutions.

For the inhomogeneous problem (1.6) without gradient term (i.e. for $b=0$), it was proved in [4] (see also [35] for the case of positive solutions) that the Fujita exponent becomes different from the homogeneous case and is given by $p^{*}:=\frac{n}{n-2}$ ($n\geq 2$). Namely, it was shown that

• $\bullet$

If $p<p^{*}$ and $\int_{\mathbb{R}^{n}}h(x)\,dx>0$, then the problem admits no global solutions;

• $\bullet$

If $n\geq 3$ and $p>p^{*}$, then for any $\delta>0$, there exists $\varepsilon>0$ such that the problem has global solutions provided that $|h(x)|,|u_{0}(x)|\leq\frac{\varepsilon}{(1+|x|^{n+\delta})}$ regardless of whether or not $\int_{\mathbb{R}^{n}}h(x)\,dx>0$.

Moreover, it was proved that when $p=\frac{n}{n-2}$ ($n\geq 3$) and under a decay condition on $h(x)$ as $|x|\to\infty$, the problem admits no global solutions.

As far as we know, the study of Fujita-type results for problems (1.3) and (1.6) was not considered previously. Note that a similar question was studied in the case of nonlinear wave equations of the type

[TABLE]

where $p,q>1$ (see [36]).

The rest of the paper is organized as follows. In Section 2, problem (1.3) is investigated. We first obtain an optimal Fujita-type result for positive solutions. Next, we discuss the case of possibly sign-changing solutions under a nonnegative initial mean condition $\int_{\mathbb{R}^{n}}u_{0}(x)\,dx\geq 0$, and derive a finite time blow-up result. In Section 3, we investigate the case of possibly sign-changing solutions to the nonhomogeneous problem (1.6) under the condition $\int_{\mathbb{R}^{n}}h(x)\,dx>0$ and obtain a Fujita-type result, with a different critical exponent.

2. The study of problem (1.3)

We first focus our attention on positive solutions and thus consider problem (1.3) with $u_{0}\geq 0$, $u_{0}\not\equiv 0$. Our first main result is given by the following theorem.

Theorem 1**.**

Let $n\geq 1$, $p>1$, $q\geq 1$ and $b>0$.

• (i)

Assume

[TABLE]

where $p_{F}$ is given by (1.11). Then for any $u_{0}\in BC^{1}(\mathbb{R}^{n})$ such that $u_{0}\geq 0$ and $u_{0}\not\equiv 0$, the solution to problem (1.3) blows up in finite time.

• (ii)

Assume

[TABLE]

Then problem (1.3) admits positive global classical solutions for suitably small initial data. Namely, this is for instance the case if $u_{0}\in BC^{1}(\mathbb{R}^{n})$ satisfies

[TABLE]

with $\varepsilon>0$ small enough.

Remark 2.1**.**

(i) In view of the results recalled in introduction, the finite time blow-up cannot be produced by the gradient term alone and should be considered as an effect of the source term $|u|^{p}$. However our result reveals the occurence of an interesting phenomenon of discontinuity of the Fujita exponent in the presence of the gradient term. Namely, as long as $q>1+\frac{1}{n+1}$, the Fujita exponent remains the same as for the problem without gradient term, i.e. $p=1+\frac{2}{n}$, and when $q$ becomes equal to, or smaller than, $1+\frac{1}{n+1}$, the Fujita exponent jumps to $p=\infty$ (i.e. no positive global solutions exist whatever the value of $p>1$). A similar situation will be encountered in the next section for the inhomogeneous problem.

(ii) A related, though different, phenomenon was observed in [1] for positive solutions of the reaction-convection-diffusion problem

[TABLE]

where $p>1$, $q\geq 1$ and $a\neq 0$. Namely, the Fujita exponent for (2.4) was shown to be given by:

[TABLE]

A first change of critical exponent thus occurs when $q$ decreases from $\frac{n+1}{n}$, but in a continuous way, and a discontinuous jump of the critical exponent then occurs when $q$ reaches $1$ from above. We refer to [12, 24] for general survey articles on critical Fujita exponents and to [10, 27, 13] for other non-standard behaviors regarding such exponents (appearing in certain time-nonlocal problems).

The following proposition will be used in the proofs of Theorems 1 and 3.

Proposition 2.1**.**

Consider problem (1.6) with $n\geq 1$, $p>1$, $q\geq 1$, $b\in\mathbb{R}$, $u_{0},h\in BC^{1}(\mathbb{R}^{n})$ and $h\geq 0$.

• (i)

Assume that

[TABLE]

for some $M,\tau>0$. Then there exists a positive constant $N=N(p,M,\tau,\|u_{0}\|_{C^{1}},\|\nabla h\|_{\infty})$ such that

[TABLE]

• (ii)

If $T<\infty$, then we have

[TABLE]

Proof.

(i) By the maximum principle, we have

[TABLE]

hence

[TABLE]

If $q\leq 2$, the assertion then follows from standard gradient bounds (see e.g. [22, 25]). Let us thus consider the case $q>2$ (but the following argument actually works for any $q>1$). Let $\sigma\in\{-1,1\}$ and denote

[TABLE]

It can be easily seen that for all $i\in\{1,\cdots,n\}$, $z_{i}$ solves

[TABLE]

where $\cdot$ denotes the scalar product in $\mathbb{R}^{n}$. Hence, using (2.5), for all $i\in\{1,\cdots,n\}$, there holds

[TABLE]

where $K:=p\max(M^{p-1},|m|^{p-1})$. On the other hand, observe that for all $i\in\{1,\cdots,n\}$,

[TABLE]

is a supersolution to (2.7). Hence, one deduces that

[TABLE]

which yields the desired gradient bound (2.6).

(ii) This is an immediate consequence of (1.7) and assertion (i). ∎

Now, we are able to prove Theorem 1.

Proof of Theorem 1..

(i) The case $p\leq 1+\frac{2}{n}$ is immediate. Indeed it is well known that the solution $v>0$ to the problem

[TABLE]

blows up in finite time in $L^{\infty}$ norm (cf. the introduction). Since, by the comparison principle, we have $u\geq v$ as long as both solutions exist, it follows that $T<\infty$.

Let us thus assume $1\leq q\leq 1+\frac{1}{n+1}$. Consider the corresponding diffusive Hamilton-Jacobi equation

[TABLE]

We know that problem (2.11) admits a unique global classical solution. By [15, Theorems 2 and 3], since $1\leq q\leq 1+\frac{1}{n+1}$, there exists $\ell\in(0,\infty)$ such that

[TABLE]

Now, we argue by contradiction by supposing that $u$ exists globally. By the comparison principle (see e.g. [30, Section 52]), it follows that

[TABLE]

Let $R>0$ to be chosen below. By (2.12) and (2.13), there exists $t_{0}=t_{0}(R)>0$ such that

[TABLE]

where we denote $B_{\rho}:=\left\{x\in\mathbb{R}^{n}:\,|x|<\rho\right\}$ for all $\rho>0$. From this lower bound, we shall reach a contradiction by using the classical Kaplan eigenfunction method [18] combined with a rescaling argument. Let $\lambda>0$ be the first eigenvalue of $-\Delta$ in $H_{0}^{1}(B_{1})$ and let $\varphi>0$ be the corresponding eigenfunction normalized by $\int_{B_{1}}\varphi\,dx=1$. Set

[TABLE]

Multiplying (1.3) by $\varphi_{R}$, integrating by parts over $B_{R}$, and using $\Delta\varphi_{R}=-\lambda R^{-2}\varphi_{R}$, for all $t>0$, we obtain

[TABLE]

where $\partial_{\nu}$ denotes the outward normal derivative on $\partial B_{R}$. Since $b>0$, $\varphi_{R}=0$ and $\partial_{\nu}\varphi_{R}\leq 0$ on $\partial B_{R}$, one deduces that

[TABLE]

Next, using $\int_{B_{R}}\varphi_{R}\,dx=1$ and Jensen’s inequality, it follows that $y(t):=\int_{B_{R}}u(t)\varphi_{R}\,dx$ satisfies

[TABLE]

But, on the other hand, (2.14) implies

[TABLE]

Choosing $R>\sqrt{\lambda}\left(\frac{\ell}{2}\right)^{\frac{1-p}{2}}$, from (2.16), we obtain $y^{p-1}(t_{0})>\lambda R^{-2}$, which implies that the right-hand side of (2.15) at $t=t_{0}$ is positive. But the differential inequality (2.15) then cannot have a global solution for $t>t_{0}$: a contradiction. We conclude that $T<\infty$.

(ii) Let

[TABLE]

with $\varepsilon,k>0$ to be chosen and $\phi(t,x)=(t+1)^{-\frac{n}{2}}\exp[{-{|x|^{2}\over 4(t+1)}}]$. Set $\mathcal{P}z:=z_{t}-\Delta z-z^{p}-b|\nabla z|^{q}$. Since $\phi$ solves the heat equation, we have

[TABLE]

Using $\phi^{p-1}\leq(t+1)^{-{n(p-1)\over 2}}$ and

[TABLE]

it follows that

[TABLE]

Now, owing to (2.1), we have ${n(p-1)\over 2}>1$ and ${n(q-1)+q\over 2}>1$. Therefore, we may choose $0<k<{n\over 2}$ small enough so that

[TABLE]

Then taking $\varepsilon>0$ sufficiently small, it follows that $\mathcal{P}z\geq 0$ in $(0,\infty)\times\mathbb{R}^{n}$. If $0\leq u_{0}(x)\leq\varepsilon e^{-\frac{|x|^{2}}{4}}$, it then follows from the comparison principle (see e.g. [30, Section 52]) that

[TABLE]

Finally, by Proposition 2.1, one deduces that $T=\infty$. This completes the proof of Theorem 1. ∎

Remark 2.2**.**

For $p,q,u_{0}$ as in Theorem 1(ii), with a little more work (at the level of the proof of Proposition 2.1), one can show that $|\nabla u|$ is actually bounded.

Now, we study problem (1.3) under the assumption $\int_{\mathbb{R}^{n}}u_{0}(x)\,dx\geq 0$. In particular, we focus our attention on sign-changing solutions. Our second main result is given by the following theorem.

Theorem 2**.**

Let $n\geq 1$, $p,q>1$ and $b>0$. Assume

[TABLE]

Then for any $u_{0}\in BC^{1}(\mathbb{R}^{n})\cap L^{1}(\mathbb{R}^{n})$ such that $\int_{\mathbb{R}^{n}}u_{0}(x)\,dx\geq 0$ and $u_{0}\not\equiv 0$, the solution to problem (1.3) blows up in finite time.

Remark 2.3**.**

(i) Assume $p>p_{F}$. We note that the assumption $(q-1)(np-1)\leq 1$ amounts to

[TABLE]

so that the assumption on $q$ in Theorem 2 is more restrictive than in the case of positive solutions in Theorem 1(i). It is an open question whether there exist global-sign changing solutions to problem (1.3) with $\int_{\mathbb{R}^{n}}u_{0}(x)\,dx\geq 0$ for q\in\bigl{(}1+\frac{1}{np-1},1+\frac{1}{n+1}\bigr{]}.

(ii) The assumption $u_{0}\in L^{1}(\mathbb{R}^{n})$ in Theorem 2 may be replaced with $(u_{0})_{-}\in L^{1}(\mathbb{R}^{n})$, and the conclusion then remains true when $\int_{\mathbb{R}^{n}}u_{0}(x)\,dx=+\infty$.

Proof of Theorem 2..

When $p\leq p_{F}$, the result follows from [28, Section 26] (which applies to supersolutions of problem (1.10) under our assumption on $u_{0}$).

Let us thus assume $(q-1)(np-1)\leq 1$. The proof is based on a rescaled test-function argument (see [28] for a general account of these methods). Let us suppose that $u$ is a global solution to (1.3). Given $\tau>0$, we introduce the test function $\psi=\psi_{\tau}$ given by

[TABLE]

with

[TABLE]

where $\theta\in C^{\infty}(\mathbb{R}_{+})$, $\xi\in C^{\infty}(\mathbb{R}^{n})$ satisfy

[TABLE]

and $r>0$ is to be chosen. Next, multiplying (1.3) by $\psi$ and integrating over $Q_{\tau}:=(0,2\tau)\times\mathbb{R}^{n}$, we obtain

[TABLE]

Using an integration by parts, there holds

[TABLE]

which yields

[TABLE]

Further, by Hölder’s inequality and the support property of $\theta$, we obtain

[TABLE]

hence

[TABLE]

with

[TABLE]

Here and below, $C$ is a positive constant (independent of $\tau$), whose value may change from line to line. In particular, using $\varepsilon$-Young inequality (with $\varepsilon=\frac{1}{2}$), we have

[TABLE]

Similarly, we get

[TABLE]

hence

[TABLE]

with

[TABLE]

Using $\varepsilon$-Young inequality (with $\varepsilon=\frac{b}{2}$), we obtain in particular

[TABLE]

By (2.19), (2.21) and (2.23), it folows that

[TABLE]

Let us next estimate $I_{i}(\tau)$, $i=1,2$. By the properties of $\psi,g$ and the change of variables $t=\tau s$, $x=\tau^{r}y$, we deduce that

[TABLE]

hence

[TABLE]

Similarly, for $I_{2}(\tau)$, we compute

[TABLE]

hence

[TABLE]

Now, combining (2.24)–(2.26), we obtain

[TABLE]

Let us make the choice $r=\frac{p(q-1)}{q(p-1)}$. With such $r$, one has

[TABLE]

and we deduce from (2.27) that, for all $\tau>0$,

[TABLE]

First consider the case $(q-1)(np-1)<1$. Using Fatou’s lemma in the first integral and the dominated convergence theorem in the second one, recalling $u_{0}\in L^{1}(\mathbb{R}^{n})$, we may pass to the infimum limit as $\tau\to\infty$ in inequality (2.29) to obtain

[TABLE]

hence $u\equiv 0$.

Finally let us consider the case $(q-1)(np-1)=1$. Then (2.25), (2.26), (2.28) and (2.29) guarantee that $I_{1}(\tau)+I_{2}(\tau)\leq C$ for all $\tau>0$ and that

[TABLE]

Going back to (2.19), (2.20), (2.22), we thus deduce that

[TABLE]

But (2.30) along with the dominated convergence theorem imply that the RHS of (2.31) goes to [math] as $\tau\to\infty$. It then follows again that $u\equiv 0$. This completes the proof of Theorem 2. ∎

3. The study of problem (1.6)

We now consider the inhomogeneous problem (1.6). Our third main result is given by the following theorem. Here and in what follows we use the convention $n/(n-2)=\infty$ for $n=2$.

Theorem 3**.**

Let $n\geq 2$, $p,q>1$ and $b>0$.

• (i)

Assume

[TABLE]

Then for any $u_{0}\in BC^{1}(\mathbb{R}^{n})$ and $h\in BC^{1}(\mathbb{R}^{n})\cap L^{1}(\mathbb{R}^{n})$ such that $\int_{\mathbb{R}^{n}}h(x)\,dx>0$, the solution to problem (1.6) blows up in finite time.

• (ii)

Let $n\geq 3$ and assume that

[TABLE]

Then there exists $h\in BC^{1}(\mathbb{R}^{n})$ with $h>0$ in $\mathbb{R}^{n}$, such that problem (1.6) admits positive global classical solutions for suitably small positive initial data $u_{0}\in BC^{1}(\mathbb{R}^{n})$.

Remark 3.1**.**

(i) Observe that in case (i) of Theorem 3 the solution to problem (1.6) may change sign. Also, the assumption $h\in L^{1}(\mathbb{R}^{n})$ may be replaced with $h_{-}\in L^{1}(\mathbb{R}^{n})$, and the conclusion then remains true when $\int_{\mathbb{R}^{n}}h(x)\,dx=+\infty$.

(ii) Similarly to problem (1.3), a phenomenon of discontinuity of the Fujita exponent in the presence of the gradient term occurs. Namely, as long as $q>\frac{n}{n-1}$ , the Fujita exponent remains the same as for the problem without gradient term, i.e. $p=\frac{n}{n-2}$, and when $q$ becomes smaller than $\frac{n}{n-1}$ ($n\geq 3$), the Fujita exponent jumps to $p=\infty$.

Proof of Theorem 3..

(i) When $p<\frac{n}{n-2}$, the result follows immediately from [4, Theorem 2.1]. Let us thus assume $q<\frac{n}{n-1}$. Let us suppose that $u$ is a global solution to (1.6). Given $\tau>0$, we introduce the test function $\varphi=\varphi_{\tau}$ given by

[TABLE]

with $g$ is given by (2.18) and

[TABLE]

where $\eta\in C^{\infty}(\mathbb{R})$ is such that

[TABLE]

Next, multiplying (1.6) by $\varphi$, integrating over $S_{\tau}:=(0,\tau)\times\mathbb{R}^{n}$, and using an integration by parts, we obtain

[TABLE]

which yields

[TABLE]

Further, following a similar argument as in the proof of Theorem 2, one obtains

[TABLE]

On the other hand, one has

[TABLE]

Hence, combining (3.2) with (3.3), there holds

[TABLE]

Now, let us make the choice $r=\frac{p(q-1)}{q(p-1)}$. With such $r$, one has

[TABLE]

and we deduce from (3.4) that, for all $\tau>0$,

[TABLE]

Since $q<\frac{n}{n-1}$ and $h\in L^{1}(\mathbb{R}^{n})$, passsing to the limit as $\tau\to\infty$ in (3.5) and using the dominated convergence theorem, we obtain

[TABLE]

which contradicts the fact that $\int_{\mathbb{R}^{n}}h(x)\,dx>0$.

(ii) We look for a supersolution of the form $v(x)=\varepsilon(1+|x|^{2})^{-k}$, where $k,\varepsilon>0$, as well as $h$, will determined below. We compute

[TABLE]

[TABLE]

Therefore,

[TABLE]

Owing to assumption (3.1), we may choose $k>0$ such that

[TABLE]

Then choosing $\varepsilon>0$ sufficiently small, we get

[TABLE]

Setting $h(x):=-\Delta v-v^{p}-b|\nabla v|^{q}>0$, we see that $v$ is a (super-)solution to the PDE in (1.6). Starting from any initial data $u_{0}\in BC^{1}(\mathbb{R}^{n})$ with $0\leq u_{0}\leq v$, we deduce from the comparison principle that $u(t,x)\leq v(x)$ in $(0,T)\times\mathbb{R}^{n}$. Applying Proposition 2.1, we conclude that $T=\infty$. ∎

Acknowledgements. The first and second authors extend their appreciation to the Deanship of Scientific Research at King Saud University for funding this work through research group No. RGP–237.