The author deals with the quasilinear parabolic equation ut = [uα + g(u)]∆u + buα+1+f(u,∇u) with Dirichlet boundary conditions in a bounded domain Ω, where f and g are lower order terms. He shows that, under suitable conditions on f and g, whether the solution is bounded or blows up in a finite time depends only on the first eigenvalue of −∆ in Ω with Dirichlet boundary condition. For some spec...

We investigate the blow-up properties of the positive solutions to a quasilinear parabolic system with nonlocal boundary condition. We first give the criteria for finite time blowup or global existence, which shows the important influence of nonlocal boundary. And then we establish the precise blow-up rate estimate. These extend the resent results of Wang et al. 2009 , which considered the spec...

We prove existence and uniqueness of solutions for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. A tipical example of energy functional we consider is the one given by the nonparametric area integrand f (x;) = p 1 + kk 2 , which corresponds with the time-dependent minimal surface equation. We also study the asimpto...

We prove existence and uniqueness of solutions for the Dirichlet problem for quasilinear parabolic equations in divergent form for which the energy functional has linear growth. A tipical example of energy functional we consider is the one given by the nonparametric area integrand f (x;) = p 1 + kk 2 , which corresponds with the time-dependent minimal surface equation. We also study the asimpto...

We study the validity of a Smoluchowski–Kramers approximation for class wave equations in bounded domain Rn subject to state-dependent damping and perturbed by multiplicative noise. prove that small mass limit solution converges stochastic quasilinear parabolic equation where noise-induced extra drift is created.

We consider the inverse problem of reconstructing the diiusion coeecient in a quasilinear parabolic diierential equation in divergence form from measurements of the solution at a nite number of points in the interior of the domain. An equation error method is developed which transforms the inverse problem into a system of linear operator equations for the diiusion coeecient, and which can be so...

Motivated by the study of conditional stability of traveling waves, we give an elementary H2 center stable manifold construction for quasilinear parabolic PDE, sidestepping apparently delicate regularity issues by the combination of a carefully chosen implicit fixed-point scheme and straightforward time-weighted Hs energy estimates. As an application, we show conditional stability of Laxor unde...

Well-posedness of a free boundary problem for electrostatic microelectromechanical systems (MEMS) is investigated when nonlinear bending effects are taken into account. The model describes the evolution of the deflection of an electrically conductive elastic membrane suspended above a fixed ground plate together with the electrostatic potential in the free domain between the membrane and the fi...

We consider two models which were both designed to describe the movement of eukaryotic cells responding to chemical signals. Besides a common standard parabolic equation for the diffusion of a chemoattractant, like chemokines or growth factors, the two models differ for the equations describing the movement of cells. The first model is based on a quasilinear hyperbolic system with damping, the ...