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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)

Citations:

Zbl 0820.34011
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References:

[1] Alt, H. W.; Luckhaus, S., Quasilinear elliptic parabolic differential equations, Math. Z., Vol. 183, 311-341 (1983) · Zbl 0497.35049
[3] Bardos, C. L.; Le Roux, A. Y.; Nedelec, J. C., First order Quasilinear Equations with boundary conditions, Comm. in partial differential equations, Vol. 4, 9, 1017-1043 (1979) · Zbl 0418.35024
[5] Bénilan, Ph., Sur des problèmes non monotones dans un espace \(L^2\), Publi. Math. Besançon, Analyse non linéaire, Vol. 3 (1977)
[7] Bénilan, Ph.; Touré, H., Sur l’équation générale \(u_t = φ(u)_{ xx } \) − \(ψ(u)_x + v\), C. R. Acad. Sc. Paris, t. 299, série I, n 18 (1984) · Zbl 0586.35016
[12] Math USSR Sbornik, Vol. 10, 217-243 (1970) · Zbl 0215.16203
[13] Kruskov, S. N.; Yu. Panov, E., Conservative quasilinear first order laws with an infinite domain of dependence on the initial data, Soviet Math. Dokl, Vol. 42, N. 2 (1991)
[14] Lions, J. L., Quelques méthodes de résolution des problèmes aux limites non linéaires (1969), Dunod-Gauthier-Villars: Dunod-Gauthier-Villars Paris · Zbl 0189.40603
[15] Oleinik, O. A., Discontinuous solutions of nonlinear differential Equations, Amer. Math. Transi., Vol. (2), 26, 95-172 (1963) · Zbl 0131.31803
[16] Pierre, M., Un Théorème général de génération de semi-groupes non linéaires, Israël Journal of Mathematics, Vol. 23, n° 3-4 (1976) · Zbl 0343.34050
[17] Touré, H., Étude des équations générales \(u_t - φ(u)_{ xx } + f(u)x = v\) par la théorie des semi-groupes non linéaires dans \(L^1\), Thèse de \(3^e\) Cycle (1982), Université de Franche-Comté
[18] Voľpert, A. I.; Hudjaev, S. I., Cauchy’s problem for degenerate second order quasilinear parabolic equations, Math. USSR-Sbornik, Vol. 7, n° 3 (1969)
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