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Source-type solutions of some quasi-regular parabolic equations. (English. Russian original) Zbl 0532.35038
Mosc. Univ. Math. Bull. 37, No. 5, 43-47 (1982); translation from Vestn. Mosk. Univ., Ser. I 1982, No. 5, 36-39 (1982).
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 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 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 \(0<m\leq \phi '(P)\leq M<\infty\).
Reviewer: N.A.Watson
35K55 Nonlinear parabolic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35A08 Fundamental solutions to PDEs
35K65 Degenerate parabolic equations