On the time decay of solutions to classes of quasilinear evolution equations
by
Gabriele Grillo
Dipartimento di Matematica, Politecnico di Torino, Torino, Italy
Coauthors: Fabio Cipriani (Politecnico di Milano)

We consider a class of quasilinear parabolic equations whose model is the nonlinear heat equation o u = \triangle_p u on a proper, open and connected domain D subset R^d, with Dirichlet boundary conditions on \partialD, supposing in addition that 2 <= p < d and that D has finite measure; here \triangle_p is the p-Laplacian operator. We prove here an analogue of the well-known time decay properties for the linear heat equation o v = \triangle v with Dirichlet boundary conditions. Namely, while in the linear case one has:

||v(t)||\infty <= c t-d/(2q)||v(0)||q

(1)

for any t > 0 and for any L^q(D) datum v(0), we prove here that in the present nonlinear case there exist \alpha, \beta, \gamma > 0, explicitly computed in terms of d, p, q such that:

||u(t)||\infty <= c m(D)\alpha t-\beta||u(0)||\gammaq

(2)

for any t > 0 and for any L^q(D) datum u(0), where m(D) is the Lebesgue measure of D, the above formula being valid, for example, for any positive (sub)solution to the equation considered. It is remarkable that the values of \alpha, \beta, \gamma involved here converge, as p --> 2, to the corresponding values appearing in the linear case.

The proof starts from the usual Sobolev inequality, which is used to prove a new family of logarithmic Sobolev inequalities (LSI). That a suitable family of LSI is equivalent, in the linear case, to bounds of the type (1), is a familiar fact in the theory of Markovian semigroups. Here it is shown how to pass from our family of LSI to bounds of the type (2) in the nonlinear case.

(T)

Date received: November 29, 1999