论文标题
一种特殊的自相似解决方案和具有强大潜力的反应扩散方程的全球解决方案
A special self-similar solution and existence of global solutions for a reaction-diffusion equation with Hardy potential
论文作者
论文摘要
Existence and uniqueness of a specific self-similar solution is established for the following reaction-diffusion equation with Hardy singular potential $$ \partial_tu=Δu^m+|x|^{-2}u^p, \qquad (x,t)\in \real^N\times(0,\infty), $$ in the range of exponents $1\leq p<m$ and dimension $ n \ geq3 $。自相似的解决方案在$ x = 0 $中不受限制，并且具有对数垂直渐近线，但是它仍然处于任何$ x \ neq0 $ and $ t \ in（0，\ infty）$中的限制，并且它是$ l^1 $ sense中的一个弱解决方案，在$ l^1 $中，它可以满足$ u（t）$（t）$（t） $ p \在[1，\ infty）$中。作为这种自相似解决方案的应用，表明存在至少存在与先前方程相关的Cauchy问题的薄弱解决方案，即任何有限的，非负和紧凑的初始条件$ U_0 $，与以前的文献相比，对关键限制$ p = m $的文献中的先前结果对比。
Existence and uniqueness of a specific self-similar solution is established for the following reaction-diffusion equation with Hardy singular potential $$ \partial_tu=Δu^m+|x|^{-2}u^p, \qquad (x,t)\in \real^N\times(0,\infty), $$ in the range of exponents $1\leq p<m$ and dimension $N\geq3$. The self-similar solution is unbounded at $x=0$ and has a logarithmic vertical asymptote, but it remains bounded at any $x\neq0$ and $t\in(0,\infty)$ and it is a weak solution in $L^1$ sense, which moreover satisfies $u(t)\in L^p(\real^N)$ for any $t>0$ and $p\in[1,\infty)$. As an application of this self-similar solution, it is shown that there exists at least a weak solution to the Cauchy problem associated to the previous equation for any bounded, nonnegative and compactly supported initial condition $u_0$, contrasting with previous results in literature for the critical limit $p=m$.
