Nonlinear Parabolic Systems Research Articles

Probabilistic Interpretation of the Cauchy Problem Solution for Systems of Nonlinear Parabolic Equations

We reduce solution of the Cauchy problem for forward and backward systems of nonlinear parabolic equations to solution of certain stochastic problem formulated in terms of stochastic differential equations. Then we state existence and uniqueness theorems for solutions of the resulting stochastic systems and deduce existence and uniqueness of the Cauchy problem solutions for the original parabolic systems. As a by-product we derive stochastic representations of solutions to the Cauchy problem for some classes of nonlinear parabolic equation systems.

Homogenization over the spatial variable in nonlinear parabolic systems

Homogenization over the spatial variable in nonlinear parabolic systems

Modeling adhesion-independent cell migration

A two-dimensional mathematical model for cells migrating without adhesion capabilities is presented and analyzed. Cells are represented by their cortex, which is modeled as an elastic curve, subject to an internal pressure force. Net polymerization or depolymerization in the cortex is modeled via local addition or removal of material, driving a cortical flow. The model takes the form of a fully nonlinear degenerate parabolic system. An existence analysis is carried out by adapting ideas from the theory of gradient flows. Numerical simulations show that these simple rules can account for the behavior observed in experiments, suggesting a possible mechanical mechanism for adhesion-independent motility.

A Second-Order Stabilization Method for Linearizing and Decoupling Nonlinear Parabolic Systems

A new time discretization method for strongly nonlinear parabolic systems is constructed by combining the fully explicit two-step backward difference formula and a second-order stabilization of wav...

The Existence Result of Renormalized Solution for Nonlinear Parabolic System with Variable Exponent and L1 Data

In this paper, we prove the existence result of a renormalized solution to a class of nonlinear parabolic systems, which has a variable exponent Laplacian term and a Leary lions operator with data belong to L1.

Observer Design for Lipschitz Nonlinear Parabolic PDE Systems With Unknown Input

In this article, a novel method to design the observer for a class of uncertain Lipschitz nonlinear parabolic partial differential equations (PDE) systems is investigated. First, the observer and the dynamic errors with undetermined parameters for the parabolic PDE systems subject to appropriate boundary conditions are presented. The conditions of the designed observer are involved. Then the analysis of asymptotic stability and $\mathcal {H}_\infty $ performance conditions for the observer design of uncertain nonlinear parabolic PDE systems are studied and presented in terms of matrix inequalities based on the Lyapunov stability theory. Finally, the effectiveness of the proposed method is validated by a numerical parabolic PDE system.

Mathematical and numerical analysis of an acid-mediated cancer invasion model with nonlinear diffusion

In this paper, we study the existence of weak solutions of the nonlinear cancer invasion parabolic system with density-dependent diffusion operators. To establish the existence result, we use regularization, the Faedo-Galerkin approximation method, some a priori estimates, and compactness arguments. Furthermore in this paper, we present results of numerical simulations for the considered invasion system with various nonlinear density-dependent diffusion operators. A standard Galerkin finite element method with the backward Euler algorithm in time is used as a numerical tool to discretize the given cancer invasion parabolic system. The theoretical results are validated by numerical examples.

Stability of Uniformly Attracting Sets for Impulsive-Perturbed Multi-Valued Semiflows

Abstract In this paper we investigate stability of uniformly attracting sets for semiflows generated by impulsive infinite-dimensional dynamical systems without uniqueness. Obtained abstract results are applied to weakly nonlinear parabolic system, whose trajectories have jumps at moments of intersection with certain surface in the phase space.

Local null controllability of coupled degenerate systems with nonlocal terms and one control force

In this paper, we are concerned with the internal control of a class of one-dimensional nonlinear parabolic systems with nonlocal and weakly degenerate diffusion coefficients. Our main theorem establishes a local null controllability result with only one internal control for a system of two equations. The proof, based on the ideias developed by Fursikov and Imanuvilov, is obtained from the global null controllability of the linearized system provided by Lyusternik's Inverse Mapping Theorem. This work extends the results previously treated by the authors for just one equation. For the system, the main issue is to obtain similar results with just one internal control, which requires a new Carleman estimate with the local term just depending on one of the state function.

Existence of renormalized solutions for nonlinear parabolic systems describing the spread of epidemics

Existence of renormalized solutions for nonlinear parabolic systems describing the spread of epidemics

Large time behavior of ODE type solutions to a nonlinear parabolic system

Let (u,v) be a solution to the Cauchy problem for a nonlinear parabolic system (P)∂tu=Δu+vαinRN×(0,∞),∂tv=Δv+uβinRN×(0,∞),u(x,0)=λ+φ(x)>0inRN,v(x,0)=μ+ψ(x)>0inRN,where α, β>0 with αβ<1, λ,μ>0 and φ,ψ∈BC(RN)∩Lr(RN) with 1≤r<∞ and infx∈RNφ(x)>−λ, infx∈RNψ(x)>−μ. Then the solution (u,v) to problem (P) behaves like a positive solution to ODE’s ζ′=ηα and η′=ζβ in (0,∞) and both of ‖u(t)‖L∞(RN) and ‖v(t)‖L∞(RN) diverge as t→∞. In this paper we obtain the precise description of the large time behavior of the solution (u,v).

On global $$L^q$$ estimates for systems with p-growth in rough domains

We study regularity results for nonlinear parabolic systems of $p$-Laplacian type with inhomogeneous boundary and initial data, with $p\in(\frac{2n}{n+2},\infty)$. We show bounds on the gradient of solutions in the Lebesgue-spaces with arbitrary large integrability exponents and natural dependences on the right hand side and the boundary data. In particular, we provide a new proof of the global non-linear Calder\'on-Zygmund theory for such systems. Our method makes use of direct estimates on the solution minus its boundary values and hence is considerably shorter than the available higher integrability results. Technically interesting is the fact that our parabolic estimates have no scaling deficit with respect to the leading order term. Moreover, in the singular case, $p\in(\frac{2n}{n+2},2]$, any scaling deficit can be omitted.

Regularity problem for one class of nonlinear parabolic systems with non-smooth in time principal matrices

Partial regularity of solutions to a class of second order nonlinear parabolic systems with non-smooth in time principal matrices is proved in the paper. The coefficients are assumed to be measurable and bounded in the time variable and VMO-smooth in the space variables uniformly with respect to time. To prove the result, we apply the so-called $A(t)$-caloric approximation method. The method was applied by the authors earlier to study regularity of quasilinear systems.

Event-triggered reliable [formula omitted] fuzzy filtering for nonlinear parabolic PDE systems with Markovian jumping sensor faults

This paper is devoted to even-triggered reliable H∞ filter design for a class of nonlinear partial differential equation (PDE) systems with Markovian jumping sensor faults. Initially, a Takagi-Sugeno (T-S) fuzzy model is adopted to reconstruct the nonlinear systems. Then, the time delays and signal quantization that often occur in network transmission are taken into account, and an integral-type event-triggered scheme is developed to improve the transmission channel utilization. Furthermore, non-parallel-distributed-compensation technique is introduced to increase the flexibility of filter design and the filter’s parameters can be obtained by solving several linear matrix inequalities. Finally, two numerical simulations and an application study to Catalytic Rod are provided to demonstrate the effectiveness and practicability of the proposed methodology.

Sampled-data fuzzy control with space-varying gains for nonlinear time-delay parabolic PDE systems

This paper introduces a sampled-data fuzzy control (SDFC) with space-varying gains for nonlinear time-delay parabolic partial differential equation (PDE) systems. First, a Takagi-Sugeno (T-S) fuzzy time-delay parabolic PDE model is employed to represent the nonlinear time-delay PDE system. Second, with the aid of the T-S fuzzy time-delay PDE model, a SDFC design with space-varying gains is developed in the formulation of space-dependent linear matrix inequalities (LMIs) by constructing an appropriate Lyapunov functional, which can stabilize exponentially the time-delay PDE system. These stabilization conditions can be applied to either slowing-varying time delay or fast-varying one. Finally, simulation results on two examples are presented to demonstrate the effectiveness of the proposed method.

Blowing Up Solutions for Nonlinear Parabolic Systems with Unequal Elliptic Operators

We give sufficient conditions for the boundedness of the blow-up set and no boundary blow-up for type I blowing up solutions to a nonlinear parabolic system with unequal elliptic operators. We introduce a new method to simplify the nonlinear parabolic system into a scalar nonlinear parabolic inequality and investigate qualitative properties of the blow-up set.

Distributed optimal control of a tumor growth treatment model with cross-diffusion effect

In this paper, we examine an optimal control problem of a coupled nonlinear parabolic system with cross-diffusion operators. The system describes the density of tumor cells, effector-immune cells, circulating lymphocyte population and chemotherapy drug concentration. The distributed control has been taken for drug concentration to control the amount of drug to be injected and to evade the side effects of the drug. We prove the existence of a weak solution of the direct problem. Then, the existence of control for the proposed control problem is proved. Further, we derive the optimality conditions and also the existence of a solution of the adjoint problem. The finite element numerical method is implemented for the proposed control problem. Then, theoretical results are illustrated with the help of numerical experiments. Finally, the importance of control function and the cross-diffusion effect are studied for the proposed control problem using numerical computations.

Homogenization of degenerate coupled transport processes in porous media with memory terms

In this paper, we establish a homogenization result for a doubly nonlinear parabolic system arising from the hygro‐thermo‐chemical processes in porous media taking into account memory phenomena. We present a mesoscale model of the composite (heterogeneous) material where each component is considered as a porous system and the voids of the skeleton are partially saturated with liquid water. It is shown that the solution of the mesoscale problem is two‐scale convergent to that of the upscaled problem as the spatial parameter goes to zero.

Second-Order Regularity for Parabolic p-Laplace Problems

Optimal second-order regularity in the space variables is established for solutions to Cauchy–Dirichlet problems for nonlinear parabolic equations and systems of p-Laplacian type, with square-integrable right-hand sides and initial data in a Sobolev space. As a consequence, generalized solutions are shown to be strong solutions. Minimal regularity on the boundary of the domain is required, though the results are new even for smooth domains. In particular, they hold in arbitrary bounded convex domains.

Asymptotic behavior of solutions to a class of coupled nonlinear parabolic systems

This paper studies the Cauchy problem to a class of coupled nonlinear parabolic systems and investigates the asymptotic behavior of solutions to the problem. The blow-up theorem of Fujita type is established by the integral estimation and suitable supersolutions. Moreover, the critical Fujita exponent determined by the diffusion and the spatial dimension is given.