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In this paper we present a parabolic approach to studying the diffusive long time behaviour of solutions to the Cauchy problem:where u0 and u1 satisfy suitable assumptions. After an appropriate scaling we obtain the convergence to a stationary solution in L q norm (1 ≤ q < ∞).2000 Mathematics Subject Classification: 35L15; 35B40.

We study a class of discrete velocity type approximations to nonlinear parabolic equations with source. After proving existence results and estimates on the solution to the relaxation system, we pass into the limit towards a weak solution, which is the unique entropy solution if the coefficients of the parabolic equation are constant.2000 Mathematics Subject Classification: 35K65; 35F25; 35L60.

The paper is devoted to the study of initial-boundary value problems for quasilinear second-order systems. Existence and uniqueness of the solution in the space , with , is proved in the case where is a half-space of . The proof of the main theorem relies on two preliminary results: existence of the solution to mixed problems for linear second-order systems with smooth coefficients, and existence of the solution to initial-boundary value problems for linear second-order operators whose coefficients depend on the variables and through a function . By means of the results proved for linear operators, the well posedness of the mixed problem for the quasi-linear system is established by studying the convergence of a suitable iteration scheme.