# zbMATH — the first resource for mathematics

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)
##### Keywords:
well-posedness; entropy solution
Full Text:
##### References:
 [1] Alt, H. W.; Luckhaus, S., Quasilinear elliptic parabolic differential equations, Math. Z., Vol. 183, 311-341, (1983) · Zbl 0497.35049 [2] H. Attouch, Variational convergence for functions and operators, Applicable Maths Series Pitmann, London, 1984. · Zbl 0561.49012 [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 [4] Ph. Bénilan, Équation d’Évolution dans un espace de Banach quelconque et application, Thèse de Doctorat d’état, Orsay, 1972. [5] Bénilan, Ph., Sur des problèmes non monotones dans un espace L^{2}, Publi. Math. Besançon, Analyse non linéaire, Vol. 3, (1977) [6] Ph. Bénilan, M. G. Crandall and A. Pazy, Evolution Equation governed by accretive Operators, (livre à paraître). [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 [8] Ph. Bénilan and H. Touré, Sur l’équation générale u_{t} = a(., u, φ(., u)_{x})_{x} dans L^{1}. I. Étude du problème stationnaire, à paraître dans Evolution Equations, Proceedings Conférence L.S.U., Janvier 1993, Marcel Dekker 1994. [9] Ph. Bénilan and R. Gariepy, Strong solutions in L^{1} of degenerate parabolic equations, à paraître dans J. Diff. Equation. [10] H. Brézis, Opérateurs maximaux monotones et semi-groupes de contraction dans les espaces de Hilbert, Math. Studies 5, North-Holland, 1973. [11] S. L. Diaz and F. de Thelin, On a nonlinear parabolic problem arising in some models related to turbulent flows, à paraître dans SIAM J. Math. Anal. · Zbl 0808.35066 [12] Kruskov, S. N., First order quasilinear equations with several independent variables, Math. Sb. 81, 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 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)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.