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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
##### MSC:
 35K55 Nonlinear parabolic equations 35D05 Existence of generalized solutions of PDE (MSC2000) 35A08 Fundamental solutions to PDEs 35K65 Degenerate parabolic equations