论文标题
对于奇特势势的准反应扩散方程的指数自相似形式的永恒溶液
Eternal solutions in exponential self-similar form for a quasilinear reaction-diffusion equation with critical singular potential
论文作者
论文摘要
我们证明了具有指数形式的$$ u（x，t）= e^{αt} f（| x | e^{ - βT}），\qquadα，\β> 0 $ 4的存在和独特性的存在和独特性。 $（x，t）\ in \ real^n \ times（0，t）$，带有$ m> 1 $，$ 1 <p <m $和$σ= -2（p-1）/（m-1）/（m-1）$，在dimension $ n \ geq2 $中，在额外的假设$ 1 <p <（m+1）中，在dimension $ n = 1 $中持有相同的结果。这种自相似的解决方案通常在文献中被称为\ emph {Eternal Solutions}，因为它们适用于（ - \ infty，\ infty）$中的任何$ t \。作为这些永恒解决方案存在的应用，我们显示了\ emph {global intime弱解决方案}的存在，任何初始条件$ u_0 \ in L^{\ infty}（\ real^n）$中的任何初始条件$ u_0 \，尤其是这些弱解决方案在任何时候保持紧凑night $ t> 0 $ t> 0 $ t> 0 $ u_0 $ compacted $ u_0 $。
We prove existence and uniqueness of self-similar solutions with exponential form $$ u(x,t)=e^{αt}f(|x|e^{-βt}), \qquad α, \ β>0 $$ to the following quasilinear reaction-diffusion equation $$ \partial_tu=Δu^m+|x|^σu^p, $$ posed for $(x,t)\in\real^N\times(0,T)$, with $m>1$, $1<p<m$ and $σ=-2(p-1)/(m-1)$ and in dimension $N\geq2$, the same results holding true in dimension $N=1$ under the extra assumption $1<p<(m+1)/2$. Such self-similar solutions are usually known in literature as \emph{eternal solutions} since they exist for any $t\in(-\infty,\infty)$. As an application of the existence of these eternal solutions, we show existence of \emph{global in time weak solutions} with any initial condition $u_0\in L^{\infty}(\real^N)$, and in particular that these weak solutions remain compactly supported at any time $t>0$ if $u_0$ is compactly supported.
