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Generalized solutions of nonlinear diffusion equations. (English) Zbl 1009.35041
The author found the generalization solution of the following Cauchy problem in the sense of Colombeau: \[ \begin{cases}\widetilde u_t+ f(x,t,\widetilde u)\widetilde u_x+ g(x, t,\widetilde u)\widetilde u= \widetilde\mu\widetilde u_{xx},\quad & x\in\mathbb{R},\;t> 0,\\ \widetilde u|_{t=0}= \widetilde u_0,\quad & x\in\mathbb{R},\end{cases} \] where \(\widetilde\mu\) is generalized constant. He proved the existence and uniqueness theorem and he showed the relationship between generalized solutions and classical solutions by using the pseudo-norm.
Reviewer: Kim Dohan (Seoul)
35K55 Nonlinear parabolic equations
35D05 Existence of generalized solutions of PDE (MSC2000)
35K15 Initial value problems for second-order parabolic equations
46F30 Generalized functions for nonlinear analysis (Rosinger, Colombeau, nonstandard, etc.)