Quasilinear Parabolic Equations Research Articles

Quasi linear parabolic pde posed on a network with non linear Neumann boundary condition at vertices

The purpose of this article is to study quasi linear parabolic partial differential equations of second order, posed on a bounded network, satisfying a nonlinear and non dynamical Neumann boundary condition at the vertices. We prove the existence and the uniqueness of a classical solution.

Solving high-dimensional forward-backward doubly SDEs and their related SPDEs through deep learning

Forward-backward doubly stochastic differential equations (FBDSDEs) are related to a type of quasi-linear parabolic stochastic partial differential equations (SPDEs). We propose a deep learning-based numerical algorithm for solving such equations. Using deep neural networks as approximations of the controls, we can deal with high-dimensional cases. Numerical experiments are carried out to demonstrate the accuracy and efficiency of the proposed numerical algorithm.

Extinction Phenomenon and Decay Estimate for a Quasilinear Parabolic Equation with a Nonlinear Source

By energy estimate approach and the method of upper and lower solutions, we give the conditions on the occurrence of the extinction and nonextinction behaviors of the solutions for a quasilinear parabolic equation with nonlinear source. Moreover, the decay estimates of the solutions are studied.

Stackelberg-Nash Controllability for a Quasi-linear Parabolic Equation in Dimension 1D, 2D, or 3D

This paper deals with the application of Stackelberg-Nash strategies to the control to quasi-linear parabolic equations in dimensions 1D, 2D, or 3D. We consider two followers, intended to solve a Nash multi-objective equilibrium; and one leader satisfying the controllability to the trajectories.

A gradient maximum principle of solutions for a quasilinear parabolic equation

This short note is concerned with a quasilinear diffusion equation under initial and Neumann boundary value condition. To be more precise, the authors establish a gradient maximum principle of classical solutions via the maximum principle and Hopf’s lemma. The result generalizes a recent work obtained by Kim (Proc Amer Math Soc 145:1203–1208, 2017).

Theoretical and numerical local null controllability of a quasi-linear parabolic equation in dimensions 2 and 3

This paper is devoted to the theoretical and numerical analysis of the null controllability of a quasi-linear parabolic equation. First, we establish a local controllability result. The proof relies on an appropriate inverse function argument. Then, we formulate an iterative algorithm for the computation of the null control and we prove a convergence result. Finally, we illustrate the analysis with some numerical experiments.

Systems of quasilinear parabolic equations in [formula omitted] and systems of quadratic backward stochastic differential equations

The objective of this paper is two-fold. The first objective is to complete the former work of Bensoussan and Frehse [2]. One big limitation of this paper was the fact that they are systems of PDE. on a bounded domain. One can expect solutions to be bounded, since one looks for smooth solutions. This is a very important property for the development of the method. It is true also that solutions which exist in a bounded domain may fail to exist on Rn, because of the lack of bounds. We give conditions so that the results of [2] can be extended to Rn. The second objective is to consider the BSDE (Backward stochastic differential equations) version of the system of PDE. This is the objective of a more recent work of Xing and Z̆itković [8]. They consider systems of BSDE with quadratic growth, which is a well-known open problem in the BSDE literature. Since the BSDE are Markovian, the problem is equivalent to the analytic one. However, because of this motivation the analytic problem is in Rn and not on a bounded domain. Xing and Z̆itković developed a probabilistic approach. The connection between the analytic problem and the BSDE is not apparent. Our objective is to show that the analytic approach can be completely translated into a probabilistic one. Nevertheless probabilistic concepts are also useful, after their conversion into the analytic framework. This is in particular true for the uniqueness result.

Quasilinear parabolic equations with first order terms and [formula omitted]-data in moving domains

The global existence of weak solutions to a class of quasilinear parabolic equations with nonlinearities depending on first order terms and integrable data in a moving domain is investigated. The class includes the p-Laplace equation as a special case. Weak solutions are shown to be global by obtaining appropriate estimates on the gradient as well as a suitable version of Aubin–Lions lemma in moving domains.

Existence and Nonexistence of the Solutions to the Cauchy Problem of Quasilinear Parabolic Equation with a Gradient Term

This paper deals with the existence and non-xistence of the solutions to the Cauchy problem of a class of quasilinear parabolic equations with a gradient term. We establish Fujita-type blowup theorems and determine the critical Fujita exponent in terms of spatial dimension, the asymptotic behavior of the coefficients of the gradient term at infinity, the exponents of spatial positions in the coefficients of the time-derivative term, and the source term. In particular, we classify the critical case as the blowup case.

Extinction in a finite time for solutions of a class of quasilinear parabolic equations

We study the property of extinction in a finite time for nonnegative solutions of 1 q ∂ ∂ t ( u q ) − ∇ ( | ∇ u | p − 2 ∇ u ) + a ( x ) u λ = 0 for the Dirichlet Boundary Conditions when q > λ > 0, p ⩾ 1 + q, p ⩾ 2, a ( x ) ⩾ 0 and Ω a bounded domain of R N ( N ⩾ 1). We prove some necessary and sufficient conditions. The threshold is for power functions when p > 1 + q while finite time extinction occurs for very flat potentials a ( x ) when p = 1 + q.

Hydrodynamics of Anisotropic Liquid Crystals in an Applied Magnetic Field

We consider a magnetic effect on hydrodynamic flows of nematic liquid crystals with unequal elastic constants in dimension two. Under a planar ansatz for the director field and velocity, we derive a reduced Ericksen--Leslie system which consists of a quasilinear parabolic equation and a Navier--Stokes equation involving critical nonlinearity due to elastic anisotropy. When unequal elastic constants are sufficiently close, we establish the global existence of smooth solutions to the reduced system and show convergence to an equilibrium as $t$ tends to $\infty$. A magnetic field-induced instability of the aligned state under the hydrodynamic flow is discussed.

Existence of solution and asymptotic behavior for a class of parabolic equations

<p style='text-indent:20px;'>We prove existence and uniqueness of a positive solution for a class of quasilinear parabolic equations. We also show some maximum principles on the derivatives of the solution and study the asymptotic behavior of the solution near the maximal time of existence.

Nonlinear parabolic equation having nonstandard growth condition with respect to the gradient and variable exponent

We are concerned with the existence of solutions to a class of quasilinear parabolic equations having critical growth nonlinearity with respect to the gradient and variable exponent. Using Schaeffer's fixed point theorem combined with the sub- and supersolution method, we prove the existence results of a weak solutions to the considered problems.

Regularity of Wong-Zakai approximation for non-autonomous stochastic quasi-linear parabolic equation on &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;$ {\mathbb{R}}^N $&lt;/tex-math&gt;&lt;/inline-formula&gt;

&lt;p style='text-indent:20px;'&gt;In this paper, we investigate a non-autonomous stochastic quasi-linear parabolic equation driven by multiplicative white noise by a Wong-Zakai approximation technique. The convergence of the solutions of quasi-linear parabolic equations driven by a family of processes with stationary increment to that of stochastic differential equation with white noise is obtained in the topology of &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;$ L^2( {\mathbb{R}}^N) $&lt;/tex-math&gt;&lt;/inline-formula&gt; space. We establish the Wong-Zakai approximations of solutions in &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;$ L^l( {\mathbb{R}}^N) $&lt;/tex-math&gt;&lt;/inline-formula&gt; for arbitrary &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;$ l\geq q $&lt;/tex-math&gt;&lt;/inline-formula&gt; in the sense of upper semi-continuity of their random attractors, where &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;$ q $&lt;/tex-math&gt;&lt;/inline-formula&gt; is the growth exponent of the nonlinearity. The &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;$ L^l $&lt;/tex-math&gt;&lt;/inline-formula&gt;-pre-compactness of attractors is proved by using the truncation estimate in &lt;inline-formula&gt;&lt;tex-math id="M7"&gt;$ L^q $&lt;/tex-math&gt;&lt;/inline-formula&gt; and the higher-order bound of solutions.&lt;/p&gt;

Correction to: Existence and Nonexistence of the Solutions to the Cauchy Problem of Quasilinear Parabolic Equation with a Gradient Term

Erratum to “Existence and nonexistence of the solutions to the Cauchy problem of quasilinear parabolic equation with a gradient term” by Mingjun Zhou and Yan Leng, 61(1):123–142, January, 2021

Expanding solutions of quasilinear parabolic equations

<p style='text-indent:20px;'>By using the theory of maximal <inline-formula><tex-math id="M1">\begin{document}$ L^{q} $\end{document}</tex-math></inline-formula>-regularity and methods of singular analysis, we show a Taylor's type expansion–with respect to the geodesic distance around an arbitrary point–for solutions of quasilinear parabolic equations on closed manifolds. The powers of the expansion are determined explicitly by the local geometry, whose reflection to the solutions is established through the local space asymptotics.

On the equivalence of pathwise mild and weak solutions for quasilinear SPDEs

The main goal of this work is to relate weak and pathwise mild solutions for parabolic quasilinear stochastic partial differential equations (SPDEs). Extending in a suitable way techniques from the theory of nonautonomous semilinear SPDEs to the quasilinear case, we prove the equivalence of these two solution concepts.

Numerical Solution of a Quasilinear Parabolic Equation with a Fractional Time Derivative

A homogeneous Dirichlet initial-boundary value problem for a quasilinear parabolic equation with Caputo fractional time derivative is considered. The coefficients of the elliptic part of the equation depend on the derivatives of the solution and satisfy the conditions providing strong monotonicity and Lipschitz-continuity of the corresponding operator. The equation is approximated by two finite-difference schemes: implicit and fractional step scheme. The stability of these finite difference schemes is proved and accuracy estimates are obtained under the condition of sufficient smoothness of the input data and the solution of the differential problem. A number of iterative methods for implementing the constructed nonlinear mesh problems are analyzed. The convergence and convergence rate of the iterative methods are substantiated. The results of numerical experiments confirming the theoretical conclusions are presented.

Reconstruction of the parameter in parabolic partial differential equations using Haar wavelet method

PurposeIn the past few years, Haar wavelet-based numerical methods have been applied successfully to solve linear and nonlinear partial differential equations. This study aims to propose a wavelet collocation method based on Haar wavelets to identify a parameter in parabolic partial differential equations (PDEs). As Haar wavelet is defined in a very simple way, implementation of the Haar wavelet method becomes easier than the other numerical methods such as finite element method and spectral method. The computational time taken by this method is very less because Haar matrices and Haar integral matrices are stored once and used for each iteration. In the case of Haar wavelet method, Dirichlet boundary conditions are incorporated automatically. Apart from this property, Haar wavelets are compactly supported orthonormal functions. These properties lead to a huge reduction in the computational cost of the method.Design/methodology/approachThe aim of this paper is to reconstruct the source control parameter arises in quasilinear parabolic partial differential equation using Haar wavelet-based numerical method. Haar wavelets possess various properties, for example, compact support, orthonormality and closed form expression. The main difficulty with the Haar wavelet is its discontinuity. Therefore, this paper cannot directly use the Haar wavelet to solve partial differential equations. To handle this difficulty, this paper represents the highest-order derivative in terms of Haar wavelet series and using successive integration this study obtains the required term appearing in the problem. Taylor series expansion is used to obtain the second-order partial derivatives at collocation points.FindingsAn efficient and accurate numerical method based on Haar wavelet has been proposed for parameter identification in quasilinear parabolic partial differential equations. Numerical results are obtained from the proposed method and compared with the existing results obtained from various finite difference methods including Saulyev method. It is shown that the proposed method is superior than the conventional finite difference methods including Saulyev method in terms of accuracy and CPU time. Convergence analysis is presented to show the accuracy of the proposed method. An efficient algorithm is proposed to find the wavelet coefficients at target time.Originality/valueThe outcome of the paper would have a valuable role in the scientific community for several reasons. In the current scenario, the parabolic inverse problem has emerged as very important problem because of its application in many diverse fields such as tomography, chemical diffusion, thermoelectricity and control theory. In this paper, higher-order derivative is represented in terms of Haar wavelet series. In other words, we represent the solution in multiscale framework. This would enable us to understand the solution at various resolution levels. In the case of Haar wavelet, this paper can achieve a very good accuracy at very less resolution levels, which ultimately leads to huge reduction in the computational cost.

A family of quasi-variable meshes high-resolution compact operator scheme for Burger's-Huxley, and Burger's-Fisher equation

We describe a quasi-variable meshes implicit compact finite-difference discretization having an accuracy of order four in the spatial direction and second-order in the temporal direction for obtaining numerical solution values of generalized Burger’s-Huxley and Burger’s-Fisher equations. The new difference scheme is derived for a general one-dimension quasi-linear parabolic partial differential equation on a quasi-variable meshes network to the extent that the magnitude of local truncation error of the high-order compact scheme remains unchanged in case of uniform meshes network. Practically, quasi-variable meshes high-order compact schemes yield more precise solution compared with uniform meshes high-order schemes of the same magnitude. A detailed exposition of the new scheme has been introduced and discussed the Fourier analysis based stability theory. The computational results with generalized Burger’s-Huxley equation and Burger’s-Fisher equation are obtained using quasi-variable meshes high-order compact scheme and compared with a numerical solution using uniform meshes high-order schemes to demonstrate capability and accuracy.