## Résolution d’équations d’évolution quasilinéaires en dimension N d’espace à l’aide d’équations linéaires en dimension $$N+1$$.(French)Zbl 0549.35055

Let A and B be two mappings from R into R and $$R^ N$$ respectively. Consider the nonlinear evolution equation $$(1)\quad u_ t+div B(u)- \Delta A(u)=0, u(x,t)\in R, x\in R^ N$$, $$t>0$$ to which the following linear equation is associated: $$(2)\quad s_ t+B'(w)\text{grad}_ x s- A'(w)\Delta_ xs=0, s(x,w,t)\in R, x\in R^ N$$, $$w\in R$$, $$t>0$$. The author establishes a relationship between the semigroup $$\{$$ S(t); $$t>0\}$$ corresponding to equation (1) and the semigroup $$\{G(t);t>0\}$$ associated to the linear equation (2); $$S(t)=\lim_{n\to\infty }(iG(t/n)j)^ n,$$ where i,j are some appropriate mappings. So the author extends his previous result concerning the hyperbolic case $$A\equiv 0$$ [see C. R. Acad. Sci., Paris, Sér. I, 292, 563-566 (1981; Zbl 0459.35006)].
Reviewer: G.Moroşanu

### MSC:

 35G10 Initial value problems for linear higher-order PDEs 35K25 Higher-order parabolic equations 35G30 Boundary value problems for nonlinear higher-order PDEs 47H20 Semigroups of nonlinear operators 47D03 Groups and semigroups of linear operators

Zbl 0459.35006
Full Text:

### References:

 [1] Bellman, R, Methods of nonlinear analysis, (1973), Academic Press New York [2] Benton, S.H, The Hamilton-Jacobi equation, (1977), Academic Press New York · Zbl 0215.28503 [3] Brenier, Y, Une application de la symétrisation de Steiner aux équations hyperboliques: la méthode de transport et écroulement, C. R. acad. sci. Paris sér. I, 292, 563-566, (1981) · Zbl 0459.35006 [4] {\scY. Brenier}, à paraître. [5] Brézis, H; Crandall, M.G, Uniqueness of solutions of the initial-value problem for ut − δφ(u) = 0, J. math. pures appl., 58, 153-163, (1979) · Zbl 0408.35054 [6] Crandall, M.G; Majda, A, Monotone difference approximations for scalar conservation laws, math. comp., 34, 1-21, (1981) · Zbl 0423.65052 [7] Fleming, W.H; Rishel, R, An integral formula for total gradient variation, Arch. math. (basel), 11, 218-222, (1960) · Zbl 0094.26301 [8] Hopf, E, On the right weak solution of the Cauchy problem for a quasilinear equation of first order, J. math. mech., 19, 483-487, (1969) · Zbl 0188.16102 [9] Kružkov, S.N, First order quasilinear with several space variables, Math. USSR-sb., 10, 217-243, (1970) · Zbl 0215.16203 [10] {\scP. L. Lions}, Generalized solutions of Hamilton-Jacobi equations, à paraître. · Zbl 0497.35001 [11] Volpert, A.I; Hudjaev, S.I, Cauchy’s problem for degenerate second order quasilinear parabolic equations, Math. USSR-sb., 7, 365-387, (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.