Generalized solutions of nonlinear diffusion equations
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.
Cauchy problem in the sense of Colombeau
scientific article; zbMATH DE number 1790846