Semilinear Degenerate Parabolic Equations Research Articles

Hierarchical null controllability of a semilinear degenerate parabolic equation with a gradient term

ABSTRACT In this paper, we apply a hierarchical strategy to a semilinear weakly degenerate parabolic equation with non-linearity that depends on the solution of the system and the spatial derivative of the solution. We use the Stackelberg–Nash strategy with one leader aiming to drive the solution to zero and two followers intended to solve a Nash equilibrium corresponding to a bi-objective optimal control problem. Since the system is semilinear, the functionals are not convex in general. To overcome this difficulty, we first prove the existence and uniqueness of the Nash quasi-equilibrium, which is a weaker formulation of the Nash equilibrium. Next, with additional conditions, we establish the equivalence between the concepts of Nash quasi-equilibrium and Nash equilibrium. We establish a suitable Carleman inequality for the adjoint system and then an observability inequality. Based on this observability inequality, we prove the null controllability of the linearized system. Then, using Kakutani's fixed point Theorem, we demonstrate the null controllability of the main system.

Dynamics of Non-Autonomous Stochastic Semi-Linear Degenerate Parabolic Equations with Nonlinear Noise

In the present paper, we aim to study the long-time behavior of a stochastic semi-linear degenerate parabolic equation on a bounded or unbounded domain and driven by a nonlinear noise. Since the theory of pathwise random dynamical systems cannot be applied directly to the equation with nonlinear noise, we first establish the existence of weak pullback mean random attractors for the equation by applying the theory of mean-square random dynamical systems; then, we prove the existence of (pathwise) pullback random attractors for the Wong–Zakai approximate system of the equation. In addition, we establish the upper semicontinuity of pullback random attractors for the Wong–Zakai approximate system of the equation under consideration driven by a linear multiplicative noise.

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>

Global attractors for a class of semilinear degenerate parabolic equations

Abstract In this paper, we consider the long-time behavior for a class of semi-linear degenerate parabolic equations with the nonlinearity f f satisfying the polynomial growth of arbitrary p − 1 p-1 order. We establish some new estimates, i.e., asymptotic higher-order integrability for the difference of the solutions near the initial time. As an application, we obtain the ( L 2 ( Ω ) , L p ( Ω ) ) \left({L}^{2}\left(\Omega ),{L}^{p}\left(\Omega )) -global attractors immediately; moreover, such an attractor can attract every bounded subset of L 2 ( Ω ) {L}^{2}\left(\Omega ) with the L p + δ {L}^{p+\delta } -norm for any δ ∈ [ 0 , + ∞ ) \delta \in \left[0,+\infty ) .

On the existence with exponential decay and the blow-up of solutions for coupled systems of semi-linear corner-degenerate parabolic equations with singular potentials

In this article, we study the initial boundary value problem of coupled semi-linear degenerate parabolic equations with a singular potential term on manifolds with corner singularities. Firstly, we introduce the corner type weighted p-Sobolev spaces and the weighted corner type Sobolev inequality, the Poincare inequality, and the Hardy inequality. Then, by using the potential well method and the inequality mentioned above, we obtain an existence theorem of global solutions with exponential decay and show the blow-up in finite time of solutions for both cases with low initial energy and critical initial energy. Significantly, the relation between the above two phenomena is derived as a sharp condition. Moreover, we show that the global existence also holds for the case of a potential well family.

Approximate Controllability of the System Governed by Double Coupled Semilinear Degenerate Parabolic Equations

Approximate Controllability of the System Governed by Double Coupled Semilinear Degenerate Parabolic Equations

Null controllability of a semilinear degenerate parabolic equation with a gradient term

This paper concerns the null controllability of a semilinear control system governed by degenerate parabolic equation with a gradient term, where the nonlinearity of the problem is involved with the first derivative. We first establish the well-posedness and prove the approximate null controllability of the linearized system, then we can get the approximate null controllability of the semilinear control system by a fixed point argument. Finally, the semilinear control system with a gradient term is shown to be null controllable.

Random attractors for stochastic semilinear degenerate parabolic equations with delay

In this paper, we consider a stochastic semilinear degenerate parabolic equation with delay in a bounded domain in RN and the nonlinearity satisfying an arbitrary polynomial growth condition. The random dynamical system generated by the equation is shown to have a random attractor in C([−τ,0],Lp(O)∩D01(O,σ)), which is a compact and invariant tempered set and attracts every tempered random subset of C([−τ,0],L2(O)) in the topology of C([−τ,0],Lp(O)). In a particular case, the random attractor consists of singleton sets (i.e., a random fixed point), which generates an exponentially stable non-trivial stationary solution. This theoretical result improves some recent ones for stochastic semilinear degenerate parabolic equations.

Stochastic semi-linear degenerate parabolic model with multiplicative noise and deterministic non-autonomous forcing

In this article, we study the dynamics of the stochastic semi-linear degenerate parabolic equations with multiplicative noise and deterministic non-autonomous forcing. Under the weaker assumption that is a general function depending on time, by using some new estimates, we discuss the -continuity with respect to initial data, as well as the attraction in higher integrability Lr-type space (r > 2) and regular space .

Stackelberg–Nash null controllability for some linear and semilinear degenerate parabolic equations

This paper deals with the application of Stackelberg–Nash strategies to the null controllability of degenerate parabolic equations. We assume that we can act on the system through a hierarchy of controls. A first control (the leader) is assumed to determine the policy; then, a Nash equilibrium pair (corresponding to a noncooperative multi-objective optimization strategy) is found; this governs the action of other controls (the followers). This way, the state of the system is driven to zero and, consequently, we solve a hierarchical null controllability problem. The main novelty in this paper is that the physical systems are governed by linear or semilinear 1D heat equations with degenerate coefficients.

Pathwise upper semi-continuity of random pullback attractors along the time axis

The pullback attractor of a non-autonomous random dynamical system is a time-indexed family of random sets, typically having the form {At(⋅)}t∈R with each At(⋅) a random set. This paper is concerned with the nature of such time-dependence. It is shown that the upper semi-continuity of the mapping t↦At(ω) for each ω fixed has an equivalence relationship with the uniform compactness of the local union ∪s∈IAs(ω), where I⊂R is compact. Applied to a semi-linear degenerate parabolic equation with additive noise and a wave equation with multiplicative noise we show that, in order to prove the above locally uniform compactness and upper semi-continuity, no additional conditions are required, in which sense the two properties appear to be general properties satisfied by a large number of real models.

Random dynamics of non-autonomous semi-linear degenerate parabolic equations on <inline-formula><tex-math id="M1">\begin{document} $\mathbb{R}^N$ \end{document}</tex-math></inline-formula> driven by an unbounded additive noise

In this paper, we study the dynamics of a non-autonomous semi-linear degenerate parabolic equation on \begin{document} $\mathbb{R}^N$ \end{document} driven by an unbounded additive noise. The nonlinearity has \begin{document} $(p,q)$ \end{document} -exponent growth and the degeneracy means that the diffusion coefficient $σ$ is unbounded and allowed to vanish at some points. Firstly we prove the existence of pullback attractor in \begin{document} $L^2(\mathbb{R}^N)$ \end{document} by using a compact embedding of the weighted Sobolev space. Secondly we establish the higher-attraction of the pullback attractor in \begin{document} $L^δ(\mathbb{R}^N)$ \end{document} , which implies that the cocycle is absorbing in \begin{document} $L^δ(\mathbb{R}^N)$ \end{document} after a translation by the complete orbit, for arbitrary \begin{document} $δ∈[2,∞)$ \end{document} . Thirdly we verify that the derived \begin{document} $L^2$ \end{document} -pullback attractor is in fact a compact attractor in \begin{document} $L^p(\mathbb{R}^N)\cap L^q(\mathbb{R}^N)\cap D_0^{1,2}(\mathbb{R}^N,σ)$ \end{document} , mainly by means of the estimate of difference of solutions instead of the usual truncation method.

Dynamics for a non-autonomous degenerate parabolic equation in $\mathfrak{D}_{0}^{1}(\Omega, \sigma)$

In this paper, we study the dynamics of a non-autonomous semilinear degenerate parabolic equation $u_{t}-div(\sigma(x)\nabla u)+ f(u)=g(x, t)$ defined on a bounded domain $\Omega\subset \mathbb{R}^N$ with smooth boundary. We first establish a Nash-Moser-Alikakos type a priori estimate for the difference of solutions near the initial time; Then we prove that the solution process $U(t,\tau)$ is continuous from $L^{2}(\Omega)$ to $\mathscr{D}_{0}^{1}(\Omega, \sigma)$ w.r.t. initial data; And finally show that the known $(L^{2}(\Omega), L^{2}(\Omega))$ pullback $\mathscr{D}_{\lambda}$-attractor indeed can attract in $\mathscr{D}_{0}^{1}(\Omega, \sigma)$-norm. Any differentiability on the forcing term is not required.

Pullback attractors for multi-valued non-compact random dynamical systems generated by semi-linear degenerate parabolic equations with unbounded delays

The theory of pullback attractors for multi-valued non-compact random dynamical systems and a method of asymptotic compactness based on the concepts of the Kuratowski measure of the non-compactness of a bounded set are used to prove the existence of pullback attractors for the multi-valued non-compact random dynamical systems associated with the semi-linear degenerate parabolic unbounded delay equations with both deterministic and random external terms.

Attractors for a class of semi-linear degenerate parabolic equations with critical exponent

We consider the degenerate parabolic equation $${ \partial_t u = \triangle_{\lambda}u + f(u)}$$ with Dirichlet boundary condition defined on a bounded domain $${\Omega \subset \mathbb{R}^N}$$ , where $${\triangle_{\lambda}}$$ , the so-called $${\triangle_{\lambda}}$$ -Laplacian, is a subelliptic operator of the type $$\triangle_{\lambda} :=\sum_{i=1}^N\partial_{x_i}(\lambda^2_i\partial_{x_i}),\quad \lambda= (\lambda_1(x),\ldots,\lambda_N(x)).$$ In this paper, we will establish the global existence of solutions and its corresponding attractor with critical nonlinearity $$|f(u)-f(v)| \leq c|u-v|\left(1+|u|^{\frac{4}{Q-2}}+|v|^{\frac{4}{Q-2}} \right),\quad\forall~~u,v\in \mathbb{R}.$$

A new blow-up condition for semi-linear edge degenerate parabolic equation with singular potentials

In this paper, we consider a class of semi-linear edge degenerate parabolic equation with singular potentials, which was proposed by Chen and Liu [Asymptotic stability and blow-up of solutions for semi-linear edge-degenerate parabolic equation with singular potentials. Discrete Contin. Dyn. Syst. 2016; 26:661–682.] in which the authors proved the solutions of the model blow up in finite time with low initial energy and critical initial energy. By constructing a new functional, we obtain a new blow-up condition, which demonstrates the possibility of finite time blow-up when the initial energy is larger than the critical initial energy.

Pullback attractors for a non-autonomous semilinear degenerate parabolic equation

In this paper, we consider the pullback attractors for a non-autonomous semilinear degenerate parabolic equation $u_{t}-\rom{div}(\sigma(x)\nabla u)+ f(u)=g(x, t)$ defined on a bounded domain $\Omega\subset \mathbb{R}^N$ with smooth boundary. We provide that the usual $(L^{2}(\Omega), L^{2}(\Omega))$ pullback $\mathscr{D}_{\lambda}$-attractor indeed can attract the $\mathscr{D}_{\lambda}$-class in $L^{2+\delta}(\Omega)$, where $\delta \in [0, \infty)$ can be arbitrary.

Stabilization of a class of semilinear degenerate parabolic equations by Ito noise

AbstractWe investigate the effect of Ito noise on the stability of stationary solutions to a class of semilinear degenerate parabolic equations with the nonlinearity satisfying an arbitrary polynomial growth condition. We will show that an Ito noise of sufficient intensity will stabilize the unstable stationary solution.

Existence and upper semicontinuity of random attractors for stochastic degenerate parabolic equations with multiplicative noises

The existence and the upper semicontinuity of random attractors in L2(RN) for stochastic semi-linear degenerate parabolic equations with multiplicative noises are obtained. Though the spatial domain is the entire space RN, a compact embedding is applied to get the compactness of random absorbing sets, from which follows the compactness of the system and the precompactness of the union of all perturbed random attractors.

Blow-up and nonexistence of solutions of some semilinear degenerate parabolic equations

In this paper we study a class of semilinear degenerate parabolic equations arising in mathematical finance and in the theory of diffusion processes. We show that blow-up of spatial derivatives of smooth solutions in finite time occurs to initial boundary value problems for a class of degenerate parabolic equations. Furthermore, nonexistence of nontrivial global weak solutions to initial value problems is studied by choosing a special test function. Finally, the phenomenon of blow-up is verified by a numerical experiment.