Quasilinear parabolic equations with first order terms and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e22" altimg="si3.svg"><mml:msup><mml:mrow><mml:mi>L</mml:mi></mml:mrow><mml:mrow><mml:mn>1</mml:mn></mml:mrow></mml:msup></mml:math>-data in moving domains

Math 220: First Order Scalar Quasilinear Equations

(1) a(x, y, u)ux + b(x, y, u)uy = c(x, y, u), with a, b, c at least C, given real valued functions. There is an immediate difference between semilinear and quasilinear equations at this point: since a and b depend on u, we cannot associate a vector field on R to the equation: we need to work on R at least to account for the (x, y, u) dependence. To achieve this, we proceed as follows. We consid...

Hyperbolic to Parabolic Relaxation Theory for Quasilinear First Order Systems

In this paper we study the limiting behavior of nonhomogeneous hyperbolic systems of balance laws when the relaxed equilibria are described by means of systems of parabolic type. In particular we obtain a complete theory for the 22 systems of genuinely nonlinear hyperbolic balance laws in 1-D with a strong dissipative term. A diierent method, which combines the div-curl lemma with accretive ope...

Quasilinear Parabolic Functional Evolution Equations

Based on our recent work on quasilinear parabolic evolution equations and maximal regularity we prove a general result for quasilinear evolution equations with memory. It is then applied to the study of quasilinear parabolic differential equations in weak settings. We prove that they generate Lipschitz semiflows on natural history spaces. The new feature is that delays can occur in the highest ...

An Application of Kubota-yokoyama Estimates to Quasilinear Wave Equations with Cubic Terms in Exterior Domains

Small global solutions for quasilinear wave equations are considered in three space dimensions in exterior domains. The obstacles are compact with smooth boundary and the local energy near the obstacles is assumed to decay exponentially with a possible loss of regularity. The null condition is needed to show global solutions for quadratic nonlinearities.

A Quasilinear Parabolic Equation With Inhomogeneous Density and Absorption