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


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: DOI


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