The nonlinear heat equation with high order mixed derivatives of the Dirac delta as initial values (Q2862138)

The authors establish the local existence of solutions to the nonlinear parabolic equation \(u_t=\Delta u+|u|^\alpha u\), \(0<t<T\), \(x\in {\mathbb R}^N\) with the initial value \(u(0)=K\partial_1\partial_2\cdots\partial_m\delta\), where \(1\leq m\leq N\), \(0<\alpha<2/(N+m)\) and \(\delta\) is the Dirac distribution. In particular, the authors derive the local existence of solutions for the above parabolic equation with the initial value in a high order negative Sobolev space \(H^{s,q}({\mathbb R}^N)\), \(s\leq -2\). Also, the authors obtain the existence of solutions with initial value \(u(0)=\lambda f\) where \(f\in C_0({\mathbb R}^N)\cap L^1({\mathbb R}^N)\) is antisymmetric in the \(x_1,x_2,\dots,x_m\) variables.