带权源的亚临界快扩散方程的消光和非消光解
Extinction and non-extinction profiles for the sub-critical fast diffusion equation with weighted source
摘要：
在快速扩散方程$0<m<m_c=(N-2)/N$的亚临界范围内，我们建立了具有加权源项$$ \partial_tu=Δu^m+|x|^σu^p, $$的快速扩散方程的灭绝和非灭绝自相似轮廓.我们认为$σ>0$和$\max\{p_c(σ),1\}<p<p_L(σ)$，其中$$ p_c(σ)=\frac{m(N+σ)}{N-2}, \qquad p_L(σ)=1+\frac{σ(1-m)}{2}. $$我们表明，一方面，积极的自相似解决方案在任何时候$t>0$，在形式$$ u(x,t)=t^αf(|x|t^β), \qquad f(ξ)\sim Cξ^{-(N-2)/m}, \qquad α>0, \ β>0 $$存在，提供$0<m<m_s=(N-2)/(N+2)$和$p_s(σ)=m(N+2σ+2)/(N-2)<p<p_L(σ)$。另一方面，我们证明了存在$p_0(σ)\in(p_c(σ),p_s(σ))$，使自相似的解决方案，提出有限的时间灭绝建立为$p\in(p_0(σ),p_s(σ))$和$p\in(p_s(σ),p_L(σ))$，但与配置文件$f(ξ)$具有不同的空间减少尾巴$|x|\to\infty$。我们还证明了不存在的自相似的解决方案的指数互补范围内的上述或如果$m\geq m_c$。
英文摘要：
We establish both extinction and non-extinction self-similar profiles for the following fast diffusion equation with a weighted source term $$ \partial_tu=Δu^m+|x|^σu^p, $$ posed for $(x,t)\in\real^N\times(0,\infty)$, $N\geq3$, in the sub-critical range of the fast diffusion equation $0<m<m_c=(N-2)/N$. We consider $σ>0$ and $\max\{p_c(σ),1\}<p<p_L(σ)$, where $$ p_c(σ)=\frac{m(N+σ)}{N-2}, \qquad p_L(σ)=1+\frac{σ(1-m)}{2}. $$ We show that, on the one hand, positive self-similar solutions at any time $t>0$, in the form $$ u(x,t)=t^αf(|x|t^β), \qquad f(ξ)\sim Cξ^{-(N-2)/m}, \qquad α>0, \ β>0 $$ exist, provided $0<m<m_s=(N-2)/(N+2)$ and $p_s(σ)=m(N+2σ+2)/(N-2)<p<p_L(σ)$. On the other hand, we prove that there exists $p_0(σ)\in(p_c(σ),p_s(σ))$ such that self-similar solutions presenting finite time extinction are established both for $p\in(p_0(σ),p_s(σ))$ and for $p\in(p_s(σ),p_L(σ))$, but with profiles $f(ξ)$ having different spatially decreasing tails as $|x|\to\infty$. We also prove non-existence of self-similar solutions in complementary ranges of exponents to the ones described above or if $m\geq m_c$.
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