Critical extinction exponents for a fast diffusion equation with nonlocal source and absorption

where $0<m<1$, $a,b,q,r>0$, Ω is a bounded domain in $R^N$ ($N\ge 1$) with smooth boundary ∂ Ω, and $u_0\in L^{\mathrm{\infty }}(\mathrm{\Omega })$ is a nonnegative nontrivial function.

The equation in (1.1) is a fast diffusion equation perturbed by both a nonlocal source term and an absorption term, which describes the diffusion of concentration of some Newtonian fluids or the density of some biological species (see [1, 2] and the references therein). What we are interested in here is the extinction in finite time of the nonnegative solutions $u(x,t)$ of (1.1), i.e. there exists a finite time $T>0$ such that the solution is nontrivial for $0<t<T$, but $u(x,t)\equiv 0$ for almost every $(x,t)\in \mathrm{\Omega }\times [T,\mathrm{\infty })$. In this case, T is called the extinction time. As one of the most important properties of solutions of evolutionary equations, extinction in finite time of solutions has been intensively studied by several authors (see [3–16] and the references therein). In particular, in a recent paper by Xu et al. [13], the authors investigated the extinction and non-extinction phenomena of solutions of Problem (1.1) and obtained the following result.

Theorem 1.1 (Theorems 1-3 in [13])

Suppose $N\ge 1$. If $0<r<1$ and $0<m=q<1$, then the nonnegative nontrivial weak solution of Problem (1.1) vanishes in finite time for any nonnegative initial data provided that either $|\mathrm{\Omega }|$ or a is sufficiently small; If $0<r<1$, then the nonnegative nontrivial weak solution of Problem (1.1) vanishes in finite time provided that $u_0$, $|\mathrm{\Omega }|$ or a is sufficiently small, and $q>C$ with C being a positive constant depending only on N, r and m; If $0<r<1$ and $m>q\ge r$, then the nonnegative nontrivial weak solution of Problem (1.1) vanishes in finite time for any nonnegative initial data provided that b is sufficiently large.

It can be seen from the above theorem that the extinction of nonnegative nontrivial weak solutions to Problem (1.1) occurs when the absorption term is in some sense strong. However, when the absorption term is suitably weak, whether Problem (1.1) admits non-extinction solutions or not is not answered in [13]. On the other hand, it can be seen from [9] that $q=m$ is the critical extinction exponent when there is no absorption term. An interesting problem is whether the absorption term can change the critical extinction exponent. We know from a recent paper [17] by Liu et al. that the critical extinction exponent is not changed (at least when the source is local) when $r=1$. However, when the absorption term is nonlinear, i.e. when $r\ne 1$, the problem is open.

Motivated by the works mentioned above, we investigate the critical extinction exponents for Problem (1.1) by constructing suitable super and sub-solutions and give a more complete classification of exponents and coefficients for the solutions to vanish in finite time or not.

By the strong maximum principle we know that $M,\mu >0$. Our main results are the following theorems.

Theorem 1.2 If $q>m$, then all the solutions $u(x,t)$ vanish in finite time for suitably small initial data $u_0(x)$. If $q=m$ and $a\mu <1$, then all the solutions $u(x,t)$ vanish in finite time for any nonnegative bounded initial data.

Theorem 1.3 If $r<min\{q,1\}$ or $r=q<1$ with $a|\mathrm{\Omega }|<b$, then all the solutions $u(x,t)$ of Problem (1.1) vanish in finite time for appropriately small initial data $u_0(x)$.

Theorem 1.4 If $q<min\{r,m\}$ or $r=q<m$ with $a\gamma >b$, then Problem (1.1) admits at least one non-extinction solution for any nonnegative initial data.

Here $\gamma >0$ will be given in the proof of Theorem 1.4.

Theorem 1.5 If $q=m<r$ with $a\mu >1$, then Problem (1.1) admits at least one non-extinction solution for any nonnegative initial data.

Theorem 1.6 If $q=m<1\le r$ with $a\mu =1$, then (1.1) admits at least one non-extinction solution $u(x,t)$ for any strictly positive initial data $u_0(x)$. If $q=m<r<1$ with $a\mu =1$, then Problem (1.1) admits at least one extinction solution $u(x,t)$ for any nonnegative initial data. If $q=m<r<1$ and $a|\mathrm{\Omega }|\le {\lambda }_1$, then $u(x,t)$ vanishes in the sense that ${lim}_{t\to +\mathrm{\infty }}{\parallel u(\cdot ,t)\parallel }_{m+1}=0$.

Remark 1.1 Comparing Theorems 1.2-1.6 with Theorem 1.1 we can see that our results complement those obtained in [13] since the case $r\ge 1$ is also considered in our paper. Moreover, according to Theorems 1.2, 1.5 and the first part of Theorem 1.6 it is easy to see that $q=m$ is the critical extinction exponent for Problem (1.1) when $r=1$, which is the same as the problems with local reaction terms (see [17]). However, when $r=1$ and $q=m$, for the nonlocal problem under consideration, the first eigenvalue ${\lambda }_1$ of −Δ in Ω no longer plays the same role as it does in the local case.

where $Q_T=\mathrm{\Omega }\times (0,T)$.

A function $u(x,t)$ is called a local solution of (1.1) if it is both a sub-solution and a super-solution for some $T>0$ and $u(x,t)$ is called a solution of (1.1) if it is a local solution of (1.1) in $Q_T$ for any $T>0$.

where $T>0$ may be chosen sufficiently small in such a way that there exists a solution $u_k$ of (2.2) on $Q_T$ for every $k\in \mathbb{N}$, and ${\parallel u_k\parallel }_{\mathrm{\infty }}$ is bounded independently of k. Furthermore, $1/l\le u_l\le u_k$ for $k<l$, and a super-solution (sub-solution) comparison theory holds for (2.2) (see [1, 19]).

Thus, we can choose the appropriate test function ξ as in [1, 19] to obtain $u\le u_k$. If u is a sub-solution of (1.1), the above argument shows that $u\le u_k$. Thus $U(x,t)$ is the maximal solution of (1.1), and this solution satisfies a sub-solution comparison principle.

Before proving our main results, we give a comparison principle for the solution of Problem (1.1), which is similar to Proposition 2.3 in [9] and can be proved by modifying the above arguments (see also [1, 10, 19]).

Proposition 2.1 Let u and v be a nonnegative bounded sub-solution and a nonnegative super-solution of (1.1), respectively. If either $q\ge 1$ and u is bounded from the above or $0<q<1$ and v has a positive lower bound, then $u(x,t)\le v(x,t)$ in $Q_T$ if $u_0(x)\le v_0(x)$ in Ω.

By the comparison principle for linear elliptic problem we know $\phi (x)\le \varphi (x)$ in Ω. Set ${\mu }_1={\int }_{\mathrm{\Omega }}\varphi (x)\mathrm{d}x$, $M_1={max}_{x\in {\overline{\mathrm{\Omega }}}^{\prime }}\varphi (x)$ and $\delta ={min}_{x\in \overline{\mathrm{\Omega }}}\varphi (x)$. It is well known from the strong maximum principle that $\delta >0$.

In addition, $v(x,t)\ge 0$ on $\partial \mathrm{\Omega }\times (0,T)$, for any $0<T<T^{\ast }$, and $v(x,0)\ge u_0(x)$ by the choice of A. Moreover, there exists a positive constant $C_1$ such that $v(x,t)\ge C_1$ in $Q_T$. Therefore, by applying Proposition 2.1 to (1.1) we see that $u(x,t)\le v(x,t)$ for $(x,t)\in Q_T$, which implies $u(x,T)\le v(x,T)$. The arbitrariness of $T<T^{\ast }$ and $v(x,T^{\ast })=0$ ensure that $u(x,T^{\ast })=0$. Furthermore, let $\tilde{u}(x,t)=u(x,t+T^{\ast })$, then $\tilde{u}(x,t)$ satisfies (1.1) with the initial condition $\tilde{u}(x,0)=0$. By the aforementioned proof, we see that $\tilde{u}(x,t)\le v(x,t)$ with any $A>0$. From the relation of the extinction time $T^{\ast }$ of $v(x,t)$ to A, it follows that $\tilde{u}(x,t)=0$ for any $t>0$, i.e. $u(x,t)=0$ for any $t\ge T^{\ast }$.

By the choice of k and $k_0$ it is easily verified that $a{(k{M}_1^{\frac{1}{m}})}^{q-m}\mu \le \frac{1}{2}<1$. Thus, by the results of Case (i), we can conclude that the solution $u(x,t)$ vanishes in finite time when the initial data are suitably small. The proof of this theorem is complete. □

Since $0<q<1$ and $b>a|\mathrm{\Omega }|$, we know by integrating the ODE that $g(t)$ vanishes at some finite time $T^{\ast }$. Moreover, as in the proof of Theorem 1.2, it can be verified that $g(t)$ is a super-solution of (1.1). Thus, by applying Proposition 2.1 to $u(x,t)$ and $g(t)$ for any $0<T<T^{\ast }$ we know that $u(x,t)$ also vanishes at $T^{\ast }$.

where $0<g_0<{(\frac{b}{a|\mathrm{\Omega }|})}^{\frac{1}{q-m}}$. Similar to the first case, it is well known that $g(t)$ vanishes in finite time since $r<1$ and $g(t)$ is a super-solution of (1.1) provided that $u_0(x)$ is small enough such that ${\parallel u_0\parallel }_{L^{\mathrm{\infty }}(\mathrm{\Omega })}\le g_0$. Applying Proposition 2.1 to $u(x,t)$ and $g(t)$ guarantees the finite time extinction of $u(x,t)$. This completes the proof of Theorem 1.3. □

where $C={max}_{t>0}\{{\lambda }_1{g}^{m-q}(t),b{g}^{r-q}(t)\}<+\mathrm{\infty }$. It is easy to see that (2.10) is valid for sufficiently small $\rho >0$ since $q<r\le m<1$.

Then $g(t)$ is nondecreasing and satisfies $g(t)\le {(\frac{a\gamma -b}{{\lambda }_1})}^{\frac{1}{m-q}}$. (The upper bound of $g(t)$ can be obtained by contradiction arguments and the monotonicity of $g(t)$ follows immediately as the upper bound is derived.) As in the proof of Case (i), we can construct a non-extinction sub-solution $v(x,t)=\rho g(t){\psi }^{\frac{1}{m}}(x)$ with $\rho >0$ sufficiently small.

Moreover, $v(x,t)\le w(x,t)$ by the choice of k. Therefore, by applying the monotonicity iteration process we can obtain a non-extinction solution $u(x,t)$ of (1.1) satisfying $v(x,t)\le u(x,t)\le w(x,t)$. The proof of Theorem 1.4 is complete. □

Since $a\mu >1$ and $m<r$, it is well known that $g(t)$ is nondecreasing and bounded above by ${(\frac{a\mu -1}{b{M}^{\frac{r}{m}}})}^{\frac{1}{r-m}}$. Then $v(x,t)$ is a sub-solution of (1.1) if $\rho >0$ is sufficiently small. On the other hand, the super-solution $w(x,t)$ can be chosen to be a large positive constant L satisfying $L\ge max\{{\parallel u_0\parallel }_{L^{\mathrm{\infty }}(\mathrm{\Omega })},{(\frac{a|\mathrm{\Omega }|}{b})}^{\frac{1}{r-q}},\rho M^{\frac{1}{m}}{max}_{t\ge 0}g(t)\}$. It can be observed that $(v,w)$ is a pair of sub-solution and super-solution of (1.1) satisfying $v(x,t)\le w(x,t)$. Therefore, by monotonicity iteration, we know that (1.1) admits at least one solution $u(x,t)$ such that $v(x,t)\le u(x,t)\le w(x,t)$. Since $v(x,t)>0$ in $\mathrm{\Omega }\times (0,+\mathrm{\infty })$, $u(x,t)$ cannot vanish at any finite time. The proof of Theorem 1.5 is complete. □