On the Fujita exponent for a Hardy-H\'{e}non equation with a spatial-temporal forcing term

Abstract

The purpose of this work is to analyze the wellposedness and the blow-up of solutions of the higher-order parabolic semilinear equation \[ u_t+(-\Delta)^{d}u=|x|^{\alpha}|u|^{p}+\zeta(t){\mathbf w}(x) \ \quad\mbox{for } (x,t)\in\mathbb{R}^{N}\times(0,\infty), \] where $d\in (0,1)\cup \mathbb{N}$, $p>1$, $-\alpha\in(0,\min(2d,N))$ or $\alpha\geq 0$ and $\zeta$ as well as ${\mathbf w}$ are suitable given functions. Given $p\geq \frac{N-2d\sigma+\alpha}{N-2d\sigma-2d}$ and setting $p_c=\frac{N(p-1)}{2d+\alpha}$, $\ell=\frac{N p_c}{N+2(\sigma+1)d p_c}$, we prove that for any data $u_0\in L^{p_c,\infty}(\mathbb{R}^N)$ and $\textbf{w}\in L^{\ell,\infty}(\mathbb{R}^N)$ with small norms there exists a unique global-in-time solution under the hypotheses $\zeta(t)=t^{\sigma}$, $\sigma\in (-1,0)$ and $N>2d$ in the space $C_{b}([0,\infty);L^{p_c,\infty}(\mathbb{R}^N))$. As a by-product, small Lebesgue data global existence follows and in particular, unconditional uniqueness holds in $C_{b}([0,\infty);L^{p_c}(\mathbb{R}^N))$ provided $p\in (\frac{N+\alpha}{N-2d},\infty)$. If either $m\in (-\infty,0]$ and $p\in (1,\frac{N-2dm+\alpha}{N-2dm-2d})$ or $m>0$ and $p>1$ where $\zeta(t)=O(t^m)$, $t\rightarrow\infty$ ($m\in \mathbb{R}$), then all solutions blow up under the additional condition $\int_{\mathbb{R}^N}\textbf{w}(x)\,dx>0$. As a consequence, we deduce that the corresponding Fujita critical exponent is a function of $\sigma$ and reads $p_{F}(\sigma)=\frac{N-2d\sigma+\alpha}{N-2d\sigma-2d}$ if $-1<\sigma<0$ and infinity otherwise.