Degenerate Parabolic Equations Research Articles

Stackelberg–Nash Null Controllability for a Non Linear Coupled Degenerate Parabolic Equations

The main purpose of this paper is to apply the notion of hierarchical control to a coupled degenerate non linear parabolic equations. We use the Stackelberg–Nash strategy with one leader and two followers. The followers solve a Nash equilibrium corresponding to a bi-objective optimal control problem and the leader a null controllability problem. Since the considered problem is non linear, the associated cost is non-convex. We first prove the existence, uniqueness and the characterization of the Nash quasi-equilibrium, which is a weak formulation of the Nash equilibrium because the cost associated to the non linear problem is non-convex. Next, we show that under suitable conditions, the Nash quasi-equilibrium is equivalent to the Nash equilibrium. Finally using some Carleman inequalities that we established, and the Kakutani’s fixed point Theorem, we brough the states of our system to the rest at final time T.

Kato square root problem for degenerate elliptic operators on bounded Lipschitz domains

Let n≥2, w be a Muckenhoupt A2(Rn) weight, Ω a bounded Lipschitz domain of Rn, and L:=−w−1div(A∇⋅) the degenerate elliptic operator on Ω with the Dirichlet or the Neumann boundary condition. In this article, the authors establish the following weighted Lp estimate for the Kato square root of L:‖L1/2(f)‖Lp(Ω,vw)∼‖∇f‖Lp(Ω,vw) for any f∈W01,p(Ω,vw) when L satisfies the Dirichlet boundary condition, or, for any f∈W1,p(Ω,vw) with ∫Ωf(x)dx=0 when L satisfies the Neumann boundary condition, where p is in an interval including 2, v belongs to both some Muckenhoupt weight class and the reverse Hölder class with respect to w, W01,p(Ω,vw) and W1,p(Ω,vw) denote the weighted Sobolev spaces on Ω, and the positive equivalence constants are independent of f. As a corollary, under some additional assumptions on w, via letting v:=w−1, the unweighted L2 estimate for the Kato square root of L that ‖L1/2(f)‖L2(Ω)∼‖∇f‖L2(Ω) for any f∈W01,2(Ω) when L satisfies the Dirichlet boundary condition, or, for any f∈W1,2(Ω) with ∫Ωf(x)dx=0 when L satisfies the Neumann boundary condition, are obtained. Moreover, as applications of these unweighted L2 estimates, the unweighted L2 regularity estimates for the weak solutions of the corresponding degenerate parabolic equations in Ω with the Dirichlet or the Neumann boundary condition are also established.

Global dynamical behavior of solutions for finite degenerate fourth-order parabolic equations with mean curvature nonlinearity

<abstract><p>In this work, the initial-boundary value problem for the global dynamical properties of solutions to a class of finite degenerate fourth-order parabolic equations with mean curvature nonlinearity is studied. With the help of the Nehari flow and Levine's concavity method, we establish some sharp-like threshold classifications of the initial data under sub-critical, critical and supercritical initial energy levels, that is, we describe the size of an initial data set. It requires the presumption that the initial data starting from one region of phase space have uniform global dynamical behavior, which means that the solution exists globally and decays via energy estimates that ultimately result in the solution tending to zero in the forward time. For the case in which the initial data corresponds to another region, we prove that the solutions related to these initial data are subject to blow-up phenomena in a finite time. In addition, we estimate the corresponding upper bound of the lifespan of the blow-up solution.</p></abstract>

Existence and blow-up of solutions for finitely degenerate semilinear parabolic equations with singular potentials

<abstract><p>In this article, we investigate the initial-boundary value problem for a class of finitely degenerate semilinear parabolic equations with singular potential term. By applying the Galerkin method and Banach fixed theorem we establish the local existence and uniqueness of the weak solution. On the other hand, by constructing a family of potential wells, we prove the global existence, the decay estimate and the finite time blow-up of solutions with subcritical or critical initial energy.</p></abstract>

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.

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.

基于退化抛物方程的期权漂移率反问题

基于退化抛物方程的期权漂移率反问题

Classification of nonnegative traveling wave solutions for the 1D degenerate parabolic equations

<p style='text-indent:20px;'>Traveling wave solutions for the one-dimensional degenerate parabolic equations are considered. The purpose of this paper is to classify the nonnegative traveling wave solutions including sense of weak solutions of these equations and to present their existence, information about their shape and asymptotic behavior. These are studied by applying the framework that combines Poincaré compactification and classical dynamical systems theory. We also aim to use these results to generalize the results of our previous studies. The key to this is the introduction of a transformation, which overcomes the generalization difficulties faced by these studies.</p>

Entropy Formulation for Triply Nonlinear Degenerate Elliptic-Parabolic-Hyperbolic Equation with Zero-Flux Boundary Condition

In this note, we investigated existence and uniqueness of entropy solution for triply nonlinear degenerate parabolic problem with zero-flux boundary condition. Accordingly to the case of doubly nonlinear degenerate parabolic hyperbolic equation, we propose a generalization of entropy formulation and prove existence and uniqueness result without any structure condition.

Local Calderón-Zygmund estimates for parabolic equations in weighted Lebesgue spaces

<abstract><p>We prove local Calderón-Zygmund type estimates for the gradient of weak solutions to degenerate or singular parabolic equations of $ p $-Laplacian type with $ p > \frac{2n}{n+2} $ in weighted Lebesgue spaces $ L^q_w $. We introduce a new condition on the weight $ w $ which depends on the intrinsic geometry concerned with the parabolic $ p $-Laplace problems. Our condition is weaker than the one in <sup>[<xref ref-type="bibr" rid="b13">13</xref>]</sup>, where similar estimates were obtained. In particular, in the case $ p = 2 $, it is the same as the condition of the usual parabolic $ A_q $ weight.</p></abstract>

Asymptotic behavior of solutions to coupled semilinear parabolic equations with general degenerate diffusion coefficients

<p style='text-indent:20px;'>This paper concerns the asymptotic behavior of solutions to one-dimensional coupled semilinear degenerate parabolic equations with superlinear reaction terms both in bounded and unbounded intervals. The equations are degenerate at a lateral boundary point and the diffusion coefficients are general functions. For the problem in a bounded interval, it is shown that there exist both nontrivial global solutions for small initial data and blowing-up solutions for large ones in the case that the degeneracy is not strong. Whereas in the case that the degeneracy is strong enough, the nontrivial solution must blow up in a finite time. For the problem in an unbounded interval, blowing-up theorems of Fujita type are established. If the degeneracy of the equations at the lateral boundary point is strong enough, it is shown that any nontrivial solution must blow up in a finite time. If the degeneracy of the equations at the lateral boundary point is not strong, it is proved that the critical Fujita curve is determined by the asymptotic behavior of the diffusion coefficient at infinity. Furthermore, the critical case is also considered.</p>

Controllability and observability for some forward stochastic complex degenerate/singular Ginzburg–Landau equations

This paper is addressed to establishing controllability and observability for some forward linear stochastic complex degenerate/singular Ginzburg-Landau equations. It is sufficient to establish appropriate observability inequalities for the corresponding backward and forward equations. The key is to prove the Carleman estimates of the forward and backward linear stochastic complex degenerate/singular Ginzburg-Landau operators. Compared with the existing deterministic results, it is necessary to overcome the difficulties caused by some complex coefficients and random terms. The results obtained cover those of deterministic cases and generalize those of stochastic degenerate parabolic equations. Moreover, the limit behavior of the coefficients in the equation is discussed.

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.

Unique Solvability of Direct and Inverse Problems for Degenerate Parabolic Equations in the Multidimensional Case

Unique Solvability of Direct and Inverse Problems for Degenerate Parabolic Equations in the Multidimensional Case

$ C^{1,\alpha} $ regularity for degenerate parabolic equations arising from the Heston model

$ C^{1,\alpha} $ regularity for degenerate parabolic equations arising from the Heston model

Null controllability of strongly degenerate parabolic equations

We consider linear one-dimensional strongly degenerate parabolic equations with measurable coefficients that may be degenerate or singular. Taking 0 as the point where the strong degeneracy occurs, we assume that the coefficient a = a(x) in the principal part of the parabolic equation is such that the function x → x/a(x) is in Lp(0,1) for some p > 1. After establishing some spectral estimates for the corresponding elliptic problem, we prove that the parabolic equation is null controllable in the energy space by using one boundary control.

Global attractors for a class of degenerate parabolic equations with memory

<p style='text-indent:20px;'>In this paper, we consider a class of weighted equations with memory</p><p style='text-indent:20px;'><disp-formula> <label/> <tex-math id="FE1"> \begin{document}$ \begin{equation*} \frac{\partial u}{\partial t}-{\rm div}(a(x)\nabla u)-\int_0^{\infty}k(s)\Delta u(t-s)\, ds+f(u) = g(x). \end{equation*} $\end{document} </tex-math></disp-formula></p><p style='text-indent:20px;'>The main characteristics of the model are that the equation contains a degenerated term with the weighted coefficient <inline-formula><tex-math id="M1">\begin{document}$ a(x) $\end{document}</tex-math></inline-formula> and a memory term. Under suitable assumptions on the weight function <inline-formula><tex-math id="M2">\begin{document}$ a(x) $\end{document}</tex-math></inline-formula>, we first investigate the existence and uniqueness of weak solutions to the initial-boundary-value problem for the above-mentioned equation. Next, we prove the existence of a global attractor. The results in this paper will extend some results in [<xref ref-type="bibr" rid="b14">14</xref>, <xref ref-type="bibr" rid="b16">16</xref>, <xref ref-type="bibr" rid="b17">17</xref>] to the memory cases which have not been studied before.</p>

Application of weak stabilization theory for degenerate parabolic equations in divergence form to a chemotaxis model for tumor invasion

This paper deals with the chemotaxis model for tumor invasion, \begin{document}$ \begin{align*} \begin{cases} u_t=\nabla\cdot(D(u,w)\nabla u -u\nabla v), &x\in\Omega,\ t>0,\\ v_t=\Delta v+wz,&x\in\Omega,\ t>0,\\ w_t=-wz,&x\in\Omega,\ t>0,\\ z_t=\Delta z-z+u,&x\in\Omega,\ t>0 \end{cases} \end{align*} $\end{document} under the conditions that $ (D(u, w)\nabla u -u\nabla v)\cdot \nu = \nabla v\cdot\nu = \nabla z\cdot \nu = 0 $ on $ \partial\Omega $ and that $ (u, v, w, z)|_{t = 0} = (u_0, v_0, w_0, z_0) $, where $ \Omega \subset \mathbb{R}^N $ ($ N = 1, 2, 3 $) is a bounded domain with smooth boundary $ \partial \Omega $ and $ \nu $ is the outward normal vector to $ \partial\Omega $. The function $ D $ is supposed to satisfy $ D(0, \sigma _2) = 0 $ and $ D(\sigma _1, \sigma _2)\geq \widetilde{D}(\sigma _1) $, where $ \widetilde{D}(\sigma _1) = k\sigma _1^{m-1} $ $ (\sigma _1\leq \sigma _0) $, $ \widetilde{D}(\sigma _1) = k\sigma _0^{m-1} $ $ (\sigma _1\geq \sigma _0) $ for some $ k>0 $, $ m>1 $ and $ \sigma_0>0 $, which means that the first equation is a degenerate parabolic equation. The previous paper [4] with Fujie and Ito proved global existence and boundedness of weak solutions to the same system, however, an imperfect stabilization property was established. The purpose of this paper is to apply weak stabilization theory in [8] to obtain that$ u(\cdot, t) \to \overline{u_0}\quad \text{weakly}^*\ \text{in}\ L^\infty(\Omega) \quad \text{as}\ t \to \infty, $where $ \overline{u_0}: = \frac{1}{|\Omega|}\int_\Omega u_0 $.

THE PROPERTIES OF THE VOLUME POTENTIAL FOR ONE PARABOLIC EQUATION WITH GROWING LOWEST COEFFICIENTS

The class of equations considered in the paper is a combination of two classes of equations: a degenerate parabolic equation of the Kolmogorov type and a parabolic equation with increasing coefficients in the group of younger members. Such a combination occurs in the problems of the theory of stochastic processes where, in the case of a normal Markov process, the Kolmogorov-Fokker-Planck equation has a similar form. The coefficients of this equations are constant in the group of principal terms and ones are increasing functions in the group of lowest terms. The article is devoted to the study of the properties of the volume potential, the kernel of which is the fundamental solution of the Cauchy problem for such an equation. Estimates of the fundamental solution of the Cauchy problem have a more complex structure than in the case of the classical Kolmogorov equation. These properties concern the existence of the derivatives included in the equation. They are used to establish theorems on the integral representations of solutions of the Cauchy problem and theorems on the correct solvability of the Cauchy problem in the corresponding classes of functional spaces. Such studies are carried out in this work. The obtained results are new and published for the first time.

On the classical fundamental solution of the Cauchy problem for one class of degenerate parabolic equations

On the classical fundamental solution of the Cauchy problem for one class of degenerate parabolic equations