Quasilinear Parabolic Research Articles

Existence of Weak Solutions for a Class of Quasilinear Parabolic Problems in Weighted Sobolev Space

In this paper, we investigate the existence and uniqueness of weak solutions for a new class of initial/boundary-value parabolic problems with nonlinear perturbation term in weighted Sobolev space. By building up the compact imbedding in weighted Sobolev space and extending Galerkin’s method to a new class of nonlinear problems, we drive out that there exists at least one weak solution of the nonlinear equations in the interval [0,T] for the fixed time T＞0.

Global Existence and Blow-up of Solutions to a Quasilinear Parabolic Equation with Nonlocal Source and Nonlinear Boundary Condition

This paper investigates the behavior of positive solution to the following $p$-Laplacian equation \begin{align*}u_t - (|u_x|^{p-2}u_x)_x = \int_{0}^a u^{\alpha}(\xi,t)d\xi+ku^\beta(x,t),\quad (x,t)\in[0,a]\times(0,T)\end{align*}with nonlinear boundary condition $u_x|_{x=0}=0$, $u_x|_{x=a}=u^q|_{x=a}$, where $p\geq 2$, $\alpha, \beta, k,q>0$. The authors first get the local existence result by a regularization method. Then under appropriate hypotheses, the authors establish that positive weak solution either exists globally or blow up in finite time by using comparison principle.

Existence and Asymptotic Behavior of Solutions for Quasilinear Parabolic Systems

This paper is concerned with the existence, uniqueness and asymptotic behavior of solutions for the quasilinear parabolic systems with mixed quasimonotone reaction functions endowed with Dirichlet boundary condition, in which the elliptic operators are allowed to be degenerate. By the method of the coupled upper and lower solutions and its monotone iterations, it is shown that a pair of coupled upper and lower solutions ensures that the unique positive solution exists and is globally stable if the quasisolutions are equal. Moreover, we study the asymptotic behavior of solutions to the Lotka-Volterra predator-prey model with the density-dependent diffusion.

Global Regularity of Weak Solutions to Quasilinear Elliptic and Parabolic Equations with Controlled Growth

We establish global regularity for weak solutions to quasilinear divergence form elliptic and parabolic equations over Lipschitz domains with controlled growth conditions on low order terms. The leading coefficients belong to the class of BMO functions with small mean oscillations with respect to x.

Systems of Quasilinear Parabolic Equations with Discontinuous Coefficients and Continuous Delays

This paper is concerned with a weakly coupled system of quasilinear parabolic equations where the coefficients are allowed to be discontinuous and the reaction functions may depend on continuous delays. By the method of upper and lower solutions and the associated monotone iterations and by difference ratios method and various estimates, we obtained the existence and uniqueness of the global piecewise classical solutions under certain conditions including mixed quasimonotone property of reaction functions. Applications are given to three 2-species Volterra-Lotka models with discontinuous coefficients and continuous delays.

A Quasilinear Parabolic System with Nonlocal Boundary Condition

We investigate the blow-up properties of the positive solutions to a quasilinear parabolic system with nonlocal boundary condition. We first give the criteria for finite time blowup or global existence, which shows the important influence of nonlocal boundary. And then we establish the precise blow-up rate estimate. These extend the resent results of Wang et al. (2009), which considered the special case , and Wang et al. (2007), which studied the single equation.

Existence and Nonexistence of Global Solutions of the Quasilinear Parabolic Equations with Inhomogeneous Terms

Existence and Nonexistence of Global Solutions of the Quasilinear Parabolic Equations with Inhomogeneous Terms

Existence and Nonexistence of Global Solutions of the Quasilinear Parabolic Equations with Inhomogeneous Terms

We consider the quasilinear parabolic equation with inhomogeneous term , , where , , , , and , . In this paper, we investigate the critical exponents of this equation.

Quasilinear Parabolic Systems with Mixed Boundary Conditions on Nonsmooth Domains

In this paper we investigate quasilinear systems of reaction-diffusion equations with mixed Dirichlet–Neumann boundary conditions on nonsmooth domains. Using techniques from maximal regularity and heat-kernel estimates we prove the existence of a unique solution to systems of this type.

Continuity of Weak Solutions for Quasilinear Parabolic Equations with Strong Degeneracy*

The aim of this paper is to study the continuity of weak solutions for quasilinear degenerate parabolic equations of the form

On the Growth Rate of Solutions of Quasilinear Parabolic Inequality at Boundary Points

Let L be a linear, uniformly elliptic, second-order operator of nondivergent type with measurable and bounded coefficients and α and K be constants, 0 0. We study the growth rate of solutions of the quasilinear parabolic inequality Lu − ut ≥ − (|∇ u|1+α + K at a boundary point.

Determination of the Right-Hand Side in a Quasilinear Parabolic Equation with a Terminal Observation

Determination of the Right-Hand Side in a Quasilinear Parabolic Equation with a Terminal Observation

Homogenization for Degenerate Quasilinear Parabolic Equations of Second Order

In this paper we study the homogenization of degenerate quasilinear parabolic equations: $$ \partial _{t} u - {\text{div}}a{\left( {\frac{t} {\varepsilon },\frac{x} {\varepsilon },u,\nabla u} \right)} = f{\left( {t,x} \right)}, $$ where a(t, y, α, λ) is periodic in (t, y).

Extremality in Solving General Quasilinear Parabolic Inclusions

The paper deals with an initial boundary-value problem for a parabolic inclusion whose multivalued term has the structure of a difference between the Clarke generalized gradient of some locally Lipschitz function verifying a unilateral growth condition and the subdifferential of a convex function, and where the elliptic part is expressed by a general quasilinear operator of the Leray-Lions type. Our results address not only the existence of solutions, but also the extremality inside an order interval determined by appropriately defined upper and lower solutions as well as the compactness of the solution set in suitable spaces.

On Doubly Degenerate Quasilinear Parabolic Equations of Higher Order

We deal with the existence of periodic solutions for doubly degenerate quasilinear parabolic equations of higher order, which can degenerate, on a part of the boundary, on a segment in the interior of the domain and in time.

A Quasilinear Parabolic Equation with Quadratic Growth of the Gradient modeling Incomplete Financial Markets

We consider a quasilinear parabolic equation with quadratic gradient terms. It arises in the modeling of an optimal portfolio which maximizes the expected utility from terminal wealth in incomplete markets consisting of risky assets and non-tradable state variables. The existence of solutions is shown by extending the monotonicity method of Frehse. Furthermore, we prove the uniqueness of weak solutions under a smallness condition on the derivatives of the covariance matrices with respect to the solution. The in influence of the non-tradable state variables on the optimal value function is illustrated by a numerical example.

ON THE GLOBAL EXISTENCE OF SOLUTIONS TO QUASILINEAR PARABOLIC EQUATIONS This work was partially supported by the National Natural Science Foundation of China, the Natural Science Foundation of Guangdong Province, and the Foundation of Zhongshan University Advanced Research Center.

We prove, under quite general assumptions, the global existence of classical solutions for quasilinear parabolic equations in bounded domains with homogeneous Dirichlet boundary conditions. The same results for weakly coupled reaction-diffusion systems are also given.

Stabilization of Solutions of Certain Singular Quasilinear Parabolic Equations

In this paper, we study singular parabolic equations with “quadratic nonlinearity.” We consider questions of unique solvability, of stabilization of solutions at infinity, and of the presence of positive solutions.

Homogenization of Quasilinear Parabolic Equations in Periodic Media†

We study the homogenization of the nonlinear parabolic equation, where , a.e. (y, s). We will consider the homogenization as a viscosity process. †Dedicated to Professor Yves Meyer with our best wishes in his retirement.

Quasilinear Parabolic Problem with p(x)-Laplacian Operator by Topological Degree

Quasilinear Parabolic Problem with p(x)-Laplacian Operator by Topological Degree