Blow-up and global existence for semilinear parabolic systems with space-time forcing terms

Abstract

We investigate the local existence, finite time blow-up and global existence of sign-changing solutions to the inhomogeneous parabolic system with space-time forcing terms $$u_t-\Delta u=|v{|}^p+t^{\sigma }{w}_1(x),v_t-\Delta v=|u{|}^q+t^{\gamma }{w}_2(x),(u(0,x),v(0,x))=(u_0(x),v_0(x)),$$ where $t>0$, $x\in {\mathbb{R}}^N$, $N\ge 1$, $p,q>1$, $\sigma ,\gamma >-1$, $\sigma ,\gamma \ne 0$, $w_1,w_2\not\equiv 0$, and $u_0,v_0\in C_0({\mathbb{R}}^N)$. For the finite time blow-up, two cases are discussed under the conditions $w_i\in L^1({\mathbb{R}}^N)$ and ${\int }_{{\mathbb{R}}^N}{w}_i(x)dx>0$, $i=1,2$. Namely, if $\sigma >0$ or $\gamma >0$, we show that the (mild) solution $(u,v)$ to the considered system blows up in finite time, while if $\sigma ,\gamma \in (-1,0)$, then a finite time blow-up occurs when $\frac{N}{2}<\max \left\{\frac{(\sigma +1)(pq-1)+p+1}{pq-1},\frac{(\gamma +1)(pq-1)+q+1}{pq-1}\right\}$. Moreover, if $\frac{N}{2}\ge \max \left\{\frac{(\sigma +1)(pq-1)+p+1}{pq-1},\frac{(\gamma +1)(pq-1)+q+1}{pq-1}\right\}$, $p>\frac{\sigma }{\gamma }$ and $q>\frac{\gamma }{\sigma }$, we show that the solution is global for suitable initial values and $w_i$, $i=1,2$.