Global existence for quasilinear diffusion equations in isotropic nondivergence form

Arendt, Wolfgang; Chill, Ralph

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Série 5, Tome 9 (2010) no. 3, pp. 523-539.

We consider the quasilinear parabolic equation $u_t-\beta (t,x,u,\nabla u)\Delta u=f(t,x,u,\nabla u)$ in a cylindrical domain, together with initial-boundary conditions, where the quasilinearity operates on the diffusion coefficient of the Laplacian. Under suitable conditions we prove global existence of a solution in the energy space. Our proof depends on maximal regularity of a nonautonomous linear parabolic equation which we use to provide us with compactness in order to apply Schaefer’s fixed point theorem.

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