Solutions Of Degenerate Parabolic Equations Research Articles

The Space $$H_p$$ of Solutions of Degenerate Parabolic Equations

The Space $$H_p$$ of Solutions of Degenerate Parabolic Equations

Fundamental Solutions for Degenerate Parabolic Equations: Existence, Properties, and some Applications

Fundamental Solutions for Degenerate Parabolic Equations: Existence, Properties, and some Applications

Qualitative Properties of Solutions of Degenerate Parabolic Equations via Energy Approaches

We consider several problems for degenerate parabolic equations exhibiting nonlinearities of various kinds. For example the equations may contain superlinear sources, causing blow up of the solutions, or damping terms; the principal part of the operator is also nonlinear. We mention as unifying features the fact that the spatial domains have non-compact boundary, and the technical approach which is based on energy methods and a priori estimates. The issues investigated include existence under optimal assumptions on the data, asymptotic behavior of solutions, existence or non-existence of global in time solutions.

Regularity for entropy solutions of degenerate parabolic equations with L^m data

In this paper, we study the regularity results for entropy solutions of a class of parabolic nonlinear parabolic equations with degenerate coercivity, when the right-hand side is in Lm with m>1.

Stabilization in degenerate parabolic equations in divergence form and application to chemotaxis systems

This paper presents a stabilization result for weak solutions of degenerate parabolic equations in divergence form. More precisely, the result asserts that the global-in-time weak solution converges to the average of the initial data in some topology as time goes to infinity. It is also shown that the result can be applied to a degenerate parabolic-elliptic Keller-Segel system.

Existence of solutions of degenerate parabolic equations with inhomogeneous density and growing data on manifolds

We consider doubly non-linear degenerate parabolic equations on Riemannian manifolds of infinite volume. The equation is inhomogeneous: indeed it contains a capacitary coefficient depending on the space variable, which we assume to decay at infinity. We prove existence of solutions for initial data growing at infinity in a suitable admissible class and some related estimates. We also prove, independently, a sup bound valid in the same geometrical setting for solutions which are a priori known to have compact support; the majorization depends on the size of the support.

Fundamental solutions for degenerate parabolic equations: existence, properties and some their applications

Fundamental solutions for degenerate parabolic equations: existence, properties and some their applications

Stability/nonstability properties of renormalized/entropy solutions for degenerate parabolic equations with $$L^{1}$$/measure data

We study the possibility to give a formulation to the degenerate parabolic problems modeled by $$\begin{aligned} (\mathcal {P}_{b}^{1})\ {\left\{ \begin{array}{ll} u_{t}-\text {div}\left[ |\nabla u|^{p-2}\nabla u)\text {/}(1+|u|)^{\theta (p-1)}\right] =\mu \text { in } (0,T)\times \Omega ,\\ u(0,x)=u_{0}(x)\text { in }\Omega ,\quad u(t,x)=0\text { on }(0,T)\times \partial \Omega , \end{array}\right. } \end{aligned}$$ where $$\theta >0$$ , $$u_{0}\in L^{1}(\Omega )$$ and $$\mu $$ is a general (nonnegative) Radon measure. We also investigate the strong stability of solutions for noncoercive absorption problems whose model $$\begin{aligned}(\mathcal {P}_{b}^{2})\ {\left\{ \begin{array}{ll} u_{t}-\text {div}\left[ |\nabla u|^{p-2}\nabla u)\text {/}(1+|u|)^{\theta (p-1)}\right] +|u|^{q-1}u=f\text { in }(0,T)\times \Omega ,\\ u(0,x)=0\text { in }\Omega ,\quad u(t,x)=0\hbox { on }(0,T)\times \partial \Omega \end{array}\right. }\end{aligned}$$ where $$q>r(p-1)[1+\theta (p-1)]\text {/}(r-p)$$ and $$f\in L^{1}_{\text {loc}}(Q\backslash K)$$ with K is a compact subset of Q of zero r-capacity (or, is a measure concentrated on a set of r-capacity zero). We prove the convergence of approximate solutions $$u_{n}$$ (related to a regular approximation $$\mu _{n}$$ of $$\mu $$ ) towards a renormalized solutions u of $$(\mathcal {P}_{b}^{1})$$ , and we extend the previous known-results on the nonstability of entropy solutions for problems $$(\mathcal {P}_{b}^{2})$$ .

Convergence of a positive nonlinear DDFV scheme for degenerate parabolic equations

In this work, we carry out the convergence analysis of a positive DDFV method for approximating solutions of degenerate parabolic equations. The basic idea rests upon different approximations of the fluxes on the same interface of the control volume. Precisely, the approximated flux is split into two terms corresponding to the primal and dual normal components. Then the first term is discretized using a centered scheme whereas the second one is approximated in a non evident way by an upstream scheme. The novelty of our approach is twofold: on the one hand we prove that the resulting scheme preserves the positivity and on the other hand we establish energy estimates. Some numerical tests are presented and they show that the scheme in question turns out to be robust and efficient. The accuracy is almost of second order on general meshes when the solution is smooth.

Complete quenching phenomenon and instantaneous shrinking of support of solutions of degenerate parabolic equations with nonlinear singular absorption

AbstractThis paper deals with nonnegative solutions of the one-dimensional degenerate parabolic equations with zero homogeneous Dirichlet boundary condition. To obtain an existence result, we prove a sharp estimate for |ux|. Besides, we investigate the qualitative behaviours of nonnegative solutions such as the quenching phenomenon, and the finite speed of propagation. Our results of the Dirichlet problem are also extended to the associated Cauchy problem on the whole domain ℝ. In addition, we also consider the instantaneous shrinking of compact support of nonnegative solutions.

The extinction versus the blow-up: Global and non-global existence of solutions of source types of degenerate parabolic equations with a singular absorption

We consider nonnegative solutions of degenerate parabolic equations with a singular absorption term and a source nonlinear term:∂tu−(|ux|p−2ux)x+u−βχ{u>0}=f(u,x,t),inI×(0,T), with the homogeneous zero boundary condition on I=(x1,x2), an open bounded interval in R. Through this paper, we assume that p>2 and β∈(0,1). To show the local existence result, we prove first a sharp pointwise estimate for |ux|. One of our main goals is to analyze conditions on which local solutions can be extended to the whole time interval t∈(0,∞), the so called global solutions, or by the contrary a finite time blow-up τ0>0 arises such that limt→τ0⁡‖u(t)‖L∞(I)=+∞. Moreover, we prove that any global solution must vanish identically after a finite time if provided that either the initial data or the source term is small enough. Finally, we show that the condition f(0,x,t)=0, ∀(x,t)∈I×(0,∞) is a necessary and sufficient condition for the existence of solution of equations of this type.

Existence and uniqueness of singular solutions of $p$-Laplacian with absorption for Dirichlet boundary condition

In this paper, we consider the existence and uniqueness of singular solutions of degenerate parabolic equations with absorption for zero homogeneous Dirichlet boundary condition. Moreover, we also get some estimates of the short time behavior of singular solutions.

Numerical Analysis of a Robust Free Energy Diminishing Finite Volume Scheme for Parabolic Equations with Gradient Structure

We present a numerical method for approximating the solutions of degenerate parabolic equations with a formal gradient flow structure. The numerical method we propose preserves at the discrete level the formal gradient flow structure, allowing the use of some nonlinear test functions in the analysis. The existence of a solution to and the convergence of the scheme are proved under very general assumptions on the continuous problem (nonlinearities, anisotropy, heterogeneity) and on the mesh. Moreover, we provide numerical evidences of the efficiency and of the robustness of our approach.

Existence of solutions for degenerate parabolic equations with singular terms

In this paper we deal with parabolic problems whose simplest model is {u′−Δpu+B|∇u|pu=0in (0,T)×Ω,u(0,x)=u0(x)in Ω,u(t,x)=0on (0,T)×∂Ω, where T>0, N≥2, p>1, B>0, and u0 is a positive function in L∞(Ω) bounded away from zero.

A new kind of the solution of degenerate parabolic equation with unbounded convection term

Abstract A new kind of entropy solution of Cauchy problem of the strong degenerate parabolic equation is introduced. If u0 ∈ L∞(RN), E = {Ei} ∈ (L2(QT))N and divE ∈ L2(QT), by a modified regularization method and choosing the suitable test functions, the BV estimates are got, the existence of the entropy solution is obtained. At last, by Kruzkov bi-variables method, the stability of the solutions is obtained.

Optimal Continuous Dependence Estimates for Fractional Degenerate Parabolic Equations

We derive continuous dependence estimates for weak entropy solutions of degenerate parabolic equations with nonlinear fractional diffusion. The diffusion term involves the fractional Laplace operator, $\Delta^{\alpha/2}$ for $\alpha \in (0,2)$. Our results are quantitative and we exhibit an example for which they are optimal. We cover the dependence on the nonlinearities, and for the first time, the Lipschitz dependence on $\alpha$ in the $BV$-framework. The former estimate (dependence on nonlinearity) is robust in the sense that it is stable in the limits $\alpha \downarrow 0$ and $\alpha \uparrow 2$. In the limit $\alpha \uparrow 2$, $\Delta^{\alpha/2}$ converges to the usual Laplacian, and we show rigorously that we recover the optimal continuous dependence result of Cockburn and Gripenberg (J Differ Equ 151(2):231-251, 1999) for local degenerate parabolic equations (thus providing an alternative proof).

$L^1$ Contraction for Bounded (Nonintegrable) Solutions of Degenerate Parabolic Equations

We obtain new $L^1$ contraction results for bounded entropy solutions of Cauchy problems for degenerate parabolic equations. The equations we consider have possibly strongly degenerate local or nonlocal diffusion terms. As opposed to previous results, our results apply without any integrability assumption on the solutions. They take the form of partial Duhamel formulas and can be seen as quantitative extensions of finite speed of propagation local $L^1$ contraction results for scalar conservation laws. A key ingredient in the proofs is a new and nontrivial construction of a subsolution of a fully nonlinear (dual) equation. Consequences of our results are maximum and comparison principles, new a priori estimates, and, in the nonlocal case, new existence and uniqueness results.

Cahn–Hilliard and thin film equations with nonlinear mobility as gradient flows in weighted-Wasserstein metrics

In this paper, we establish a novel approach to proving existence of non-negative weak solutions for degenerate parabolic equations of fourth order, like the Cahn–Hilliard and certain thin film equations. The considered evolution equations are in the form of a gradient flow for a perturbed Dirichlet energy with respect to a Wasserstein-like transport metric, and weak solutions are obtained as curves of maximal slope. Our main assumption is that the mobility of the particles is a concave function of their spatial density. A qualitative difference of our approach to previous ones is that essential properties of the solution – non-negativity, conservation of the total mass and dissipation of the energy – are automatically guaranteed by the construction from minimizing movements in the energy landscape.

A note on entropy solutions for degenerate parabolic equations with L<SUP align="right">1</SUP> ∩ L<SUP align="right">p</SUP> data

We prove existence and uniqueness results for entropy solutions of degenerate parabolic equations with appropriately growing nonlinearities and unbounded initial data.

Numerical solutions of time fractional degenerate parabolic equations by variational iteration method with Jumarie-modified Riemann-Liouville derivative

In this article, the fractional variational iteration method is employed for computing the approximate analytical solutions of degenerate parabolic equations with fractional time derivative. The time-fractional derivatives are described by the use of a new approach, the so-called Jumarie modified Riemann–Liouville derivative, instead in the sense of Caputo. The approximate solutions of our model problem are calculated in the form of convergent series with easily computable components. Moreover, the numerical solution is compared with the exact solution and the quantitative estimate of accuracy is obtained. The results of the study reveal that the proposed method with modified fractional Riemann–Liouville derivatives is efficient, accurate, and convenient for solving the fractional partial differential equations in multi-dimensional spaces without using any linearization, perturbation or restrictive assumptions. Copyright © 2011 John Wiley & Sons, Ltd.