Upper and lower solutions in quasilinear parabolic boundary value problems

The authors study a class of initial boundary value problems associated with parabolic quasilinear equations: by introducing special auxiliary functions, upper and lower solutions are obtained, which turn out to be sharp in the sense that they coincide with the solution in particular situations. To Larry Payne on the occasion of his 80th birthday. Received: February 3, 2004; revised: April 26, 2004 Partially supported by University of Cagliari     

Some convergence results for quasilinear parabolic boundary value problems in cylindrical domains of large size

The goal of this paper is to study the asymptotic behavior of the solution of the quasilinear parabolic boundary value problems defined on cylindrical domains when one or several directions go to infinity. We show that the dimension of the space can be reduced and the rate of convergence is analyzed. The evolution p$p$-Laplacian equations and the generalized heat problems are considered.     

Boundary layers for compressible Navier-Stokes equations with outflow boundary condition

In this paper, we study the asymptotic relation between the solutions to the initial boundary value problem of the one-dimensional compressible full Navier-Stokes equations with outflow boundary condition and the associated Euler equations. We assume all the three characteristics to the corresponding Euler equations are all negative up to some small time, then we prove the existence and the stability of the boundary layers as long as the strength of the boundary layers is suitably small.     

Continuous dependence inequalities for a class of quasilinear parabolic problems

In this paper, we establish continuous dependence inequalities for the solutions u(x, t) of a class of nonlinear parabolic initial-boundary value problems and their gradients when the data are subject to variations. Dedicated to Joseph Hersch on the occasion of his 80th birthday (Received: February 24, 2005; revised: March 14, 2005)     

Long time behavior of classical solutions to the generalized Goursat problem of quasilinear hyperbolic systems

We investigate the existence of a global classical solution to the generalized Goursat problem. Under some degenerate assumptions of boundary conditions, we prove that the solution approaches a combination of Lipschitz continuous and a piecewise C¹ traveling wave solution.     

Positive solutions of quasilinear parabolic systems with Dirichlet boundary condition

Coupled systems for a class of quasilinear parabolic equations and the corresponding elliptic systems, including systems of parabolic and ordinary differential equations are investigated. The aim of this paper is to show the existence, uniqueness, and asymptotic behavior of time-dependent solutions. Also investigated is the existence of positive maximal and minimal solutions of the corresponding quasilinear elliptic system. The elliptic operators in both systems are allowed to be degenerate in the sense that the density-dependent diffusion coefficients Di(ui) may have the property Di(0)=0 for some or all i=1,…,N, and the boundary condition is ui=0. Using the method of upper and lower solutions, we show that a unique global classical time-dependent solution exists and converges to the maximal solution for one class of initial functions and it converges to the minimal solution for another class of initial functions; and if the maximal and minimal solutions coincide then the steady-state solution is unique and the time-dependent solution converges to the unique solution. Applications of these results are given to three model problems, including a scalar polynomial growth problem, a coupled system of polynomial growth problem, and a two component competition model in ecology.     

Stable and unstable manifolds for quasilinear parabolic systems with fully nonlinear boundary conditions

We investigate quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions on bounded or exterior domains in the setting of Sobolev–Slobodetskii spaces. We establish local wellposedness and study the time and space regularity of the solutions. Our main results concern the asymptotic behavior of the solutions in the vicinity of a hyperbolic equilibrium. In particular, the local stable and unstable manifolds are constructed. Dedicated to Giuseppe Da Prato on the occasion of his 70th birthday     

Stability estimates for parabolic problems with Wentzell boundary conditions

Of concern is the uniformly parabolic problem

     

Global classical solutions of mixed initial-boundary value problem for quasilinear hyperbolic systems

In this paper, we consider the mixed initial-boundary value problem for quasilinear hyperbolic systems with nonlinear boundary conditions in the first quadrant . Under the assumptions that the system is strictly hyperbolic and linearly degenerate or weakly linearly degenerate, the global existence and uniqueness of C¹ solutions are obtained for small initial and boundary data. We also present two applications for physical models.     

On convergence of solutions to equilibria for quasilinear parabolic problems

We show convergence of solutions to equilibria for quasilinear parabolic evolution equations in situations where the set of equilibria is non-discrete, but forms a finite-dimensional C¹-manifold which is normally hyperbolic. Our results do not depend on the presence of an appropriate Lyapunov functional as in the ?ojasiewicz-Simon approach, but are of local nature.     

Estimates for solutions of quasilinear problems with dead cores

A Rayleigh-Faber-Krahn type inequality is used to derive bounds for boundary value problems appearing in reaction-diffusion problems where the reactant is consumed. Interesting quantities are the minimum of the solution and the measure of the set where it vanishes. The proofs are rather elementary and apply to problems possessing solutions in a weak sense.     

Equidistributed error mesh for problems with exponential boundary layers

In this paper we present a new piecewise-linear finite element mesh suitable for the discretization of the one-dimensional convection–diffusion equation -εu^″-bu^′=0$-\epsilon u^{″}-{\mathit{bu}}^{\prime }=0$, u(0)=0$u(0)=0$, u(1)=1$u(1)=1$. The solution to this equation exhibits an exponential boundary layer which occurs also in more complicated convection–diffusion problems of the form -εΔu-b∂u/∂x+cu=f$-\epsilon \mathrm{\Delta }u-b\partial u/\partial x+\mathit{cu}=f$. The new mesh is based on the equidistribution of the interpolation error and it takes into account finite computer arithmetic. It is demonstrated numerically that for the above problem, the new mesh has remarkably better convergence properties than the well-known Shishkin and Bakhvalov meshes.     

Stability of parabolic problems with nonlinear Wentzell boundary conditions

Of concern is the nonlinear uniformly parabolic problem

     

Blow-up,global existence and exponential decay estimates for a class of quasilinear parabolic problems

This paper deals with a class of nonlinear parabolic problems in divergence form whose solutions, without appropriate data restrictions, might blow up at some finite time. The purpose of this paper is to establish conditions on the data sufficient to guarantee blow-up of solution at some finite time τ$\tau$, conditions to ensure that the solution remains bounded as well as conditions to derive some explicit exponential decay bounds for the solution and its derivatives.     

Existence of solutions for a class of singular quasilinear elliptic resonance problems

The paper concerns a resonance problem for a class of singular quasilinear elliptic equations in weighted Sobolev spaces. The equation set studied is one of the most useful sets of Navier-Stokes equations; these describe the motion of viscous fluid substances such as liquids, gases and so on. By using Galerkin-type techniques, the Brouwer fixed point theorem, and a new weighted compact Sobolev-type embedding theorem established by Shapiro, we show the existence of a nontrivial solution.     

Quenching for nonlinear degenerate parabolic problems

For the problem given by u_τ=(ξ^ru^mu_ξ)_ξ/ξ^r+f(u) for 0<ξ<a, 0<τ<Λ≤∞, u(ξ,0)=u₀(ξ) for 0≤ξ≤a, and u(0,τ)=0=u(a,τ) for 0<τ<Λ, where a and m are positive constants, r is a constant less than 1, f(u) is a positive function such that lim_u→c⁻f(u)=∞ for some positive constant c, and u₀(ξ) is a given function satisfying u₀(0)=0=u₀(a), this paper studies quenching of the solution u.     

Blow-up phenomena for some nonlinear parabolic problems

We consider the blow-up of solutions of equations of the form

_ut=div(ρ(|∇u|²) grad u)+f(u)$u_t=\text{div}\left(\rho \left({\left|\nabla u\right|}^2\right)\text{ grad }u\right)+f\left(u\right)$

On some global well-posedness and asymptotic results for quasilinear parabolic equations

We prove that the quasilinear parabolic initial-boundary value problem (1.1) below is globally well-posed in a class of high order Sobolev solutions, and that these solutions possess compact, regular attractors ast+.