Generalized solutions of parabolic initial value problems with discontinuous nonlinearities
The author studies generalized solutions of the initial value problem \[ \begin{gathered} u_t= u_{xx}+ f(u),\quad 0< t< T,\quad x\in\mathbb{R},\\ u|_{t=0}= u_0(x),\quad x\in\mathbb{R},\end{gathered}\tag{1} \] where \(f\) is expressed in the form \[ f(u)= g(u)- g(1) H(a-2)\tag{2} \] with \(\alpha\in (0,1)\) is a constant, \(H\) is the function on \(\mathbb{R}\) given by \[ H(u)= 0\quad\text{in }(-\infty, 0),\quad H(u)= 1\quad\text{in }(0,\infty)\quad\text{and }0\leq H(0)\leq 1\tag{3} \] and \(g\) is Lipschitz continuous function such that \(g(0)= 0\), \(g(u)< 0\) on \((0,\alpha]\) and \(g(u)- g(1)> 0\) on \([\alpha,1)\). The goal of this paper is to discuss the relationship between generalized and weak solutions for (1)--(3).
scientific article; zbMATH DE number 5072968
scientific article