Nonlinear Degenerate Parabolic Equations Research Articles

Existence of Compactly Supported Solutions for a Degenerate Nonlinear Parabolic Equation with NonLipschitz Source Term

In this paper the authors study the existence of nonnegative compactly supported solutions of a nonlinear degenerate parabolic equation with a non-Lipschitz source term in one space dimension. More precisely, they investigate the properties of nonnegative solutions of the problem {∂tu−ν∂2x(up)=uα+Ψ(u),u(0,x)=u0, x∈R, t>0, u=u(t,x),(1) where Ψ∈C∞ is an increasing function, satisfying Ψ(0)=0 (unforced case), Ψ(x)≤Cx for some C>0 and every x≥0. Equation (1) mimics the properties of the classical k-e system-model in the context of turbulent mixing flows with respect to nonlinearities and support properties of solutions. The authors highlight that the originality of the method resides in the fact that they deal with a non-Lipschitz source term, and in the comparison of not only the speed but also the acceleration of the boundary of the compact support. Here p>1 so that the diffusion is strongly degenerate and α>0 with 2−p≤α<1, so the nonlinearity is not Lipschitz. Equation (1) is a crude simplification of a k-e system of the form ∂tk+∂x(ku)+e=∂x(Cμk2e∂xk)+Pk2e, ∂te+∂x(eu)+C1e2k=∂x(Cμσek2e∂xe)+C2Pk, where k and e denote the specific turbulent kinetic energy and its dissipation rate, u is the velocity field, and P is proportional to turbulence production terms. The present work is focused on the lack of Lipschitz continuity of the source term and the degeneracy of the diffusion. Actually, the study is restricted to the scalar case, as in the k-l turbulence model, where l=k3/2/e is constant. Nonlinear parabolic equations similar to (1) appear also in various applications, most frequently to describe phenomena of thermal propagation in an absorptive medium. Note that such kinds of equations have been studied for slow or fast diffusion and strong or small absorption; see [R. Ferreira and J. L. Vazquez, Nonlinear Anal. 43 (2001), no. 8, Ser. A: Theory Methods, 943-985; MR1812069 (2002f:35141); R. Ferreira, V. A. Galaktionov and J. L. Vazquez, Nonlinear Anal. 50 (2002), no. 4, Ser. A: Theory Methods, 495-507; MR1923525 (2003h:35109); V. A. Galaktionov, S. I. Shmarev and J. L. Vazquez, Arch. Ration. Mech. Anal. 149 (1999), no. 3, 183-212; MR1726675 (2001k:35167)] and the references therein. Finally, the authors prove the following result: Theorem. Assume that u0 is a smooth nonnegative function with support of the form [α,β] and that the derivatives of up0 at α, β do not vanish. Then there exists a smooth, nonnegative, compactly supported solution u to (1) defined for all positive times.

Some notes on a nonlinear degenerate parabolic equation

In this note we study nonexistence and long time behavior of solutions for a nonlinear degenerate parabolic equation of non-divergence type. In addition, we also construct some special explicit solutions which may blow up in finite time.

On the Cauchy problem for a doubly nonlinear degenerate parabolic equation with strongly nonlinear sources

In this article, we consider the existence of local and global solution to the Cauchy problem of a doubly nonlinear equation. By introducing the norms |||f||| h and 〈f〉h, we give the sufficient and necessary conditions on the initial value to the existence of local solution of doubly nonlinear equation. Moreover some results on the global existence and nonexistence of solutions are considered.

Potential Symmetry Properties of a Family of Equations Occuring in Ice Sheet Dynamics

In this paper we derive some similarity solutions of a nonlinear equation associated with a free boundary problem arising in the shallow-water approximation in glaciology. In addition we present a classical potential symmetry analysis of this second-order nonlinear degenerate parabolic equation related to non-Newtonian ice sheet dynamics in the isothermal case. After obtaining a general result connecting the thickness function of the ice sheet and the solution of the nonlinear equation (without any unilateral formulation), a particular example of a similarity solution to a problem formulated with Cauchy boundary conditions is described. This allows us to obtain several qualitative properties on the free moving boundary in the presence of an accumulation-ablation function with realistic physical properties.

Mathematical Analysis of a Model of River Channel Formation

The study of overland flow of water over an erodible sediment leads to a coupled model describing the evolution of the topographic elevation and the depth of the overland water film. The spatially uniform solution of this model is unstable, and this instability corresponds to the formation of rills, which in reality then grow and coalesce to form large-scale river channels. In this paper we consider the deduction and mathematical analysis of a deterministic model describing river channel formation and the evolution of its depth. The model involves a degenerate nonlinear parabolic equation (satisfied on the interior of the support of the solution) with a super-linear source term and a prescribed constant mass. We propose here a global formulation of the problem (formulated in the whole space, beyond the support of the solution) which allows us to show the existence of a solution and leads to a suitable numerical scheme for its approximation. A particular novelty of the model is that the evolving channel self-determines its own width, without the need to pose any extra conditions at the channel margin.

Blow-up for doubly degenerate nonlinear parabolic equations

In this paper, we give a complete picture of the blow-up criteria for weak solutions of the Dirichlet problem of some doubly degenerate nonlinear parabolic equations.

Nonlinear degenerate parabolic equations with singular coefficients for Greiner vector fields

In this work, we consider the following nonlinear parabolic partial differential equations: in a cylinder Ω × (0, T) with initial condition u(w, 0) = u 0(w) ≥ 0 and vanishing on the boundary ∂Ω × (0, T), where Ω is a Carnot Carathéodory metric ball in ℝ2n+1 with smooth boundary and the time-dependent potential function is , 0 < m < q and 1 < p < q + 1. We investigate the nonexistence of positive solutions of these two problems and present our results on nonexistence.

Conditions for instantaneous support shrinking and sharp estimates for the support of the solution of the Cauchy problem for a doubly non-linear parabolic equation with absorption

Instantaneous support shrinking is studied for a doubly non-linear degenerate parabolic equation in the case of slow diffusion when, in general, the Cauchy initial data are Radon measures. For a non-negative solution, a necessary and sufficient condition for instantaneous support shrinking is obtained in terms of the local behaviour of the mass of the initial data. In the same terms, estimates are obtained for the size of the support, that are sharp with respect to order.Bibliography: 24 titles.

Global gradient estimates for degenerate parabolic equations in nonsmooth domains

This paper studies the global regularity theory for degenerate nonlinear parabolic partial differential equations. Our objective is to show that weak solutions belong to a higher Sobolev space than assumed a priori if the complement of the domain satisfies a capacity density condition and if the boundary values are sufficiently smooth. Moreover, we derive integrability estimates for the gradient. The results extend to the parabolic systems as well. The higher integrability estimates provide a useful tool in several applications.

Global existence of solutions of a nonlinear degenerate parabolic problem

In the paper, we showed the local existence and uniqueness of weak solution of a nonlinear degenerate parabolic equation with nonlinear boundary conditions, and we also obtained global solvability conditions for the situation with and without strong absorption terms.

Boundedness and blowup for nonlinear degenerate parabolic equations

The author deals with the quasilinear parabolic equation u t = [ u α + g ( u ) ] Δ u + b u α + 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 − Δ in Ω with Dirichlet boundary condition. For some special cases, the result is sharp.

Intrinsic Harnack estimates for some doubly nonlinear degenerate parabolic equations

We prove an intrinsic Harnack inequality for non-negative local weak solutions of a wide class of doubly nonlinear degenerate parabolic equations whose prototype is \begin{equation*} u_t-\mathrm{div}(u^{m-1}|Du|^{p-2}Du)=0,\qquad p{\geqslant} 2, m{\geqslant} 1. \end{equation*} As a consequence, we get that such solutions are locally Hölder continuous.

Similarity solutions for a replicator dynamics equation

We construct a family of self-similar solutions for a nonlinear degenerate parabolic equation containing a nonlocal term. This equation serves as a replicator dynamics model where the set of strategies is a continuum.

A fourth order nonlinear degenerate parabolic equation

In this paper, we study a fourth order degenerate parabolic equation, which arises in epitaxial growth of nanoscale thin films. We establish the existence of weak solutions, based on the uniform estimates for the approximate solutions. The nonnegativity and the finite speed of propagation of perturbations of solutions are also discussed.

Mini-Workshop: Control of Free Boundaries

The field of the mathematical and numerical analysis of systems of nonlinear pdes involving interfaces and free boundaries is a burgeoning area of research. Many such systems arise from mathematical models in ma- terial science and fluid dynamics such as phase separation in alloys, crystal growth, dynamics of multiphase fluids and epitaxial growth. In applications of these mathematical models, suitable performance indices and appropriate control actions have to be specified. Mathematically this leads to optimiza- tion problems with pde constraints including free boundaries. It is now timely to consider such control problems because of the maturity of the field of com- putational free boundary problems. The aim of the mini-workshop was to bring together leading experts and young researchers from the separate fields of numerical free boundary problems and optimal control in order to estab- lish links and to identify suitable model problems to serve as paradigms for progressing knowledge of optimal control of free boundaries.

Parabolic equations with non–linear, degenerate and space–time dependent operators

In this paper existence and regularity results for a class of degenerate nonlinear parabolic equations are proved. Indeed, the diffusion operator may degenerate as the solution diverges and may depend on space and time variables in a non–regular way, too. Some estimates on the behaviour of the solutions for t → +∞ are also given.

Lubrication Approximation for Thin Viscous Films: Asymptotic Behavior of Nonnegative Solutions

We use standard regularized equations and adapted entropy functionals to prove exponential asymptotic decay in the H 1 norm for nonnegative weak solutions of fourth-order nonlinear degenerate parabolic equations of lubrication approximation for thin viscous film type. The weak solutions considered arise as limits of solutions for the regularized problems. Relaxed problems, with second-order nonlinear terms of porous media type are also successfully treated by the same means. The problems investigated here are one-dimensional in space, with power-law nonlinearities. Our approach is direct and natural, as it is adapted to deal with the more complex nonlinear terms occurring in the regularized, approximating problems.

Existence and uniqueness of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian

The existence, uniqueness and regularity of viscosity solutions to the Cauchy–Dirichlet problem are proved for a degenerate nonlinear parabolic equation of the form \(u_t = \Delta_\infty u\) , where \(\Delta_\infty\) denotes the so-called infinity-Laplacian given by \(\Delta_\infty u = \langle D^2 u Du, Du \rangle\) . To do so, a coercive regularization of the equation is introduced and barrier function arguments are also employed to verify the equi-continuity of approximate solutions. Furthermore, the Cauchy problem is also studied by using the preceding results on the Cauchy–Dirichlet problem.

Extinction and Positivity for a Doubly Nonlinear Degenerate Parabolic Equation

The aims of this paper are to discuss the extinction and positivity for the solution of the initial boundary value problem and Cauchy problem of ut = div(|∇um|p−2∇um). It is proved that the weak solution will be extinct for 1 1 + 1/m for large t, where m > 0.

A new family of higher order nonlinear degenerate parabolic equations

There has been much investigation of higher order nonlinear degenerate equations of the form where M is a specified function and H is the quadratic first order energy functional . The energy functional arises in many physical models, but is not universal among higher order parabolic equations. Recent investigations have motivated the study of other energy functionals, such as for p ≠ 2. We undertake such a study here, proving the existence of weak solutions for appropriate boundary conditions, nonnegativity and positivity properties of solutions. Moreover, an entropy dissipation–entropy estimate for solutions of this equation is obtained. Support properties and long time behaviour of solutions are also discussed for various cases.