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On the general equation \(u_ t = a(.,u,\varphi (.,u)_ x)_ x + v\) in \(L^ 1\). II: The evolution problem. (Sur l’équation générale \(u_ t = a(.,u,\varphi (.,u)_ x)_ x + v\) dans \(L^ 1\). II: Le problème d’évolution.) (French) Zbl 0839.35068
Summary: [For part I, see Lect. Notes Pure Appl. Math. 168, 35-62 (1994; Zbl 0820.34011).]
We consider the general equation \(u_t= a(., u, \varphi(.,u)_x)_x+ v\) of parabolic type, which may degenerate into first-order hyperbolic type for some values of \((x, u)\). Under very general assumptions on the data, we prove existence, uniqueness and continuous dependence results for mild solution of associated Cauchy problem or boundary value problems. With additional assumptions on the data, we show that this mild solution is an “entropy solution”. We study uniqueness of a weak solution and existence of strong solution.

MSC:
35K65 Degenerate parabolic equations
35L65 Hyperbolic conservation laws
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
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