论文标题
半线性抛物线系统具有时空强迫术语的爆炸和全球存在
Blow-up and global existence for semilinear parabolic systems with space-time forcing terms
论文作者
论文摘要
我们研究了对不均匀抛物线系统的局部存在，有限的时间爆炸和全球存在，其签名解决方案具有时空强迫术语$ _t-u_t-ΔU= | v | v |^{p}+t^σw_1（x），\，\，\，\，\，v_t-t-ΔV= | （u（0，x），v（0，x））=（u_0（x），v_0（x）），$$其中$ t> 0 $，$ x \ in \ mathbb {r}^n $，$ n \ geq 1 $，$ p，$ p，q> 1 $，$ q> 1 $，$σ，$σ $ w_1，w_2 \ not \ equiv0 $，$ u_0，v_0 \ in C_0（\ Mathbb {r}^n）$。对于有限的时间爆炸，在l^1（\ Mathbb {r}^n）$和$ \ int _ {\ Mathbb {r}^n} w_i（x）\，dx> 0 $，$ i = 1,2 $的条件下讨论了两种情况。也就是说，如果$σ> 0 $或$γ> 0 $，我们表明（温和的）解决方案$（u，v）$在有限的时间内爆炸，而如果$σ，γ\ in（-1,1,0）$，则在$ \ frac {n} {2} {2} {2} {2} {2} {2} {2} <n}时发生有限的时间爆炸\ max \ left \ {\ frac {（σ+1）（pq-1）+p+1} {pq-1}，\ frac {（γ+1）（pq-1）（pq-1）+q+1} {pq-1} {pq-1} {pq-1} \ right \} $。此外，如果$ \ \ \ frac {n} {2} \ geq \ max \ left \ left \ {\ frac {（σ+1）（PQ-1）（pq-1）+p+1} {pq-1} {pq-1}，\ frac {（γ+1）（pq-1）+q+1} $ q> \fracγσ$，我们证明该解决方案是适合初始值的全局，$ w_i $，$ i = 1,2 $。
We investigate the local existence, finite time blow-up and global existence of sign-changing solutions to the inhomogeneous parabolic system with space-time forcing terms $$ u_t-Δu =|v|^{p}+t^σw_1(x),\,\, v_t-Δv =|u|^{q}+t^γw_2(x),\,\, (u(0,x),v(0,x))=(u_0(x),v_0(x)), $$ where $t>0$, $x\in \mathbb{R}^N$, $N\geq 1$, $p,q>1$, $σ,γ>-1$, $σ,γ\neq0$, $w_1,w_2\not\equiv0$, 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 $σ>0$ or $γ>0$, we show that the (mild) solution $(u,v)$ to the considered system blows up in finite time, while if $σ,γ\in(-1,0)$, then a finite time blow-up occurs when $\frac{N}{2}< \max\left\{\frac{(σ+1)(pq-1)+p+1}{pq-1},\frac{(γ+1)(pq-1)+q+1}{pq-1}\right\}$. Moreover, if $\frac{N}{2}\geq \max\left\{\frac{(σ+1)(pq-1)+p+1}{pq-1},\frac{(γ+1)(pq-1)+q+1}{pq-1}\right\}$, $p>\fracσγ$ and $q>\fracγσ$, we show that the solution is global for suitable initial values and $w_i$, $i=1,2$.
