Abstract:

We consider the nonlinear heat equation $u_t-\Delta u=|u|^\alpha u$ on $\Bbb{R}^N$, where $\alpha>0$ and $N\ge 1$. We prove that in the range $0<\alpha<{4\over {N-2}}$, for every $\mu>0$, there exist infinitely many sign-changing, self-similar solutions to the Cauchy problem with initial value $u_0(x)=\mu|x|^{-{2\over\alpha}}$. The construction is based on the analysis of the related inverted profile equation. In particular, we construct (sign-changing) self-similar solutions for positive initial values for which it is known that there does not exist any local, nonnegative solution.

摘要：

我们考虑非线性热方程$ u_t- \ Delta u = | u | ^ \ alpha u $在$ \ Bbb {R} ^ N $上，其中$ \ alpha> 0 $和$ N \ ge 1 $。我们证明，在$ 0 <\ alpha <{4 \ over {N-2}} $的范围内，对于每个$ \ mu> 0 $，对于带有初始值的Cauchy问题，存在无限多个变号，自相似解值$ u_0（x）= \ mu | x | ^ {-{2 \ over \ alpha}} $。该构造基于相关倒置轮廓方程的分析。尤其是，我们为正初始值构造（改变符号）自相似解，因为已知其不存在任何局部非负解。