Source-type solutions of some quasi-regular parabolic equations (Q788904)

Consider the equation \(u_ t=\{\phi([\psi(u)]_ x)\}_ x\) on the half- plane \(\underset \tilde{} R\times [0,\infty [\), with the initial condition that u(.,0) is the \(\delta\)-function. In the case where \(\phi\) and \(\psi\) are power functions, the solution of this problem was given by \textit{G. I. Barenblatt} [Prikl. Mat. Mekh. 16, 679-698 (1952)]. The case where \(\phi\) is the identity and \(\psi\) is not a power was investigated by \textit{S. Kamin} [J. Math. Anal. Appl. 64, 263-276 (1978; Zbl 0387.76083]. The present author considers the case where \(\psi\) is the identity and \(\phi\) is a function which is locally in \(C^{2+\gamma}\) for \(0<\gamma \leq 1\) and satisfies \(\phi '(P)>0\) for \(P\neq 0\), \(\phi\) '(0)\(\geq 0\), \(\phi(0)=0\). He proves the existence of a weak solution under the additional condition that 0\(\leq P\phi ''(P)\leq A\phi '(P)\) for some \(A>0\), and the existence and uniqueness of a strong solution if \(0<m\leq \phi '(P)\leq M<\infty\).