non-Newtonian Filtration Equation Research Articles

Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content

Cauchy problem for a non-Newtonian filtration equation with slowly decaying volumetric moisture content

Bounds on Blow-Up Time for a Higher-Order Non-Newtonian Filtration Equation

ABSTRACT Let d ∈ {1, 2, 3, . . .} and Ω ⊂ ℝ d be open bounded with Lipschitz boundary. Consider the reaction-diffusion parabolic problem P { u t | x | 4 + Δ ( | Δ u | m − 2 Δ u ) = k ( t ) | u | p − 1 u ( x , t ) ∈ Ω × ( 0 , T ) , u ( x , t ) = ∂ x j u ( x , t ) = 0 , ( x , t ) ∈ ∂ Ω × ( 0 , T ) , j ∈ { 1 , … , d } , u ( x , 0 ) = u 0 ( x ) , x ∈ Ω , where T > 0, m ∈ [2, ∞), p ∈ (1, ∞) and 0 ≠ u 0 ∈ W 0 2 , m ( Ω ) ∩ L p + 1 ( Ω ) $0 \ne u_0 \in W^{2,m}_0(\Omega) \cap L^{p+1}(\Omega)$ . We investigate the upper and lower bounds on the blow-up time of a weak solution to (P).

Quenching estimates for a non-Newtonian filtration equation with singular boundary conditions

This study concerns with the quenching features of solutions of the non-Newtonian filtration equation. Various conditions on the initial condition are shown to guarantee quenching at either the left or right boundary. Theoretical quenching rates and lower bounds to the quenching time are determined are certain cases. Numerical experiments are provided to illustrate and provide additional validation of the theoretical predictions to the quenching rates and times.

The Nonexistence of the Solutions for the Non-Newtonian Filtration Equation with Absorption

The Nonexistence of the Solutions for the Non-Newtonian Filtration Equation with Absorption

New approaches for periodic wave solutions of a non-Newtonian filtration equation with variable delay

A type of non-Newtonian filtration equations with variable delay is considered. Using a new approach which was established by Ge and Ren in (Nonlinear Anal. 58:477–488, 2004), we obtain the existence of periodic wave solutions for the non-Newtonian filtration equations. The methods of the present paper are markedly different from the existing ones.

Periodic wave solutions of non-Newtonian filtration equations with nonlinear sources and singularities

This paper is concerned with a kind of non-Newtonian filtration equation with nonlinear sources and singularities. Based on the mountain pass theorem and variational methods, a sufficient criterion for the new results on the periodic wave solutions has been provided. Here not only the structure is more general and practical than the existing works but the conditions imposed are concise. Consequently, compared with the previous results on the singular equations and non-Newtonian filtration equations, the results we established are more generalized and some previous ones can been complemented and improved. Finally, the effectiveness of the established results are validated via two numerical examples and simulations.

Periodic wave solutions of a non-Newtonian filtration equation with an indefinite singularity

This paper is concerned with the existence of periodic wave solutions for a type of non-Newtonian filtration equations with an indefinite singularity. A sufficient criterion for the existence of periodic wave solutions for non-Newtonian filtration equation is provided via an innovative method of combining a new continuation theorem with coincidence degree theory as well as mathematical analysis skills. The novelty of the present paper is that it is the first time to discuss the existence of periodic wave solutions for the indefinite singular non-Newtonian filtration equations. Finally, two numerical examples are presented to illustrate the effectiveness and feasibility of the proposed criterion in the present paper.

Boundary layer behavior of the non-Newtonian filtration equation with a small physical parameter

We study the initial-boundary value problem for the one-dimensional non-Newtonian filtration equation with a small physical parameter. We show the existence of a boundary layer as the physical parameter goes to zero, and obtain an almost optimal thickness of the boundary layer. Moreover, an optimal thickness of the boundary layer is established for some special boundary functions.

Well-posedness of quasilinear singular diffusion equations with boundary degeneracy

This paper concerns the well-posedness problems of a class ofquasilinear singular diffusion equations with boundary degeneracy.In addition to the possibledegeneracy and singularity likethe non-Newtonian filtration equations, the equations aredegenerate on a portion of the boundary due to the diffusion coefficients.The diffusion coefficients are anisotropic.For the well-posedness problems of such equations,the boundary condition may not be prescribed on the entire lateral boundary.The prescription of the boundary condition is determined bythe degeneracy of the diffusion coefficients along the outernormal direction of the spatial domain.The well-posedness problems of the equations are formulatedand the existence and uniqueness of weak solutions to the problems are proved.

Solitary wave and periodic wave solutions for the non-Newtonian filtration equations with nonlinear sources and a time-varying delay

This paper is concerned with the non-Newtonian filtration equations with nonlinear sources and a time-varying delay. By an extension of Mawhin's continuation theorem and some analysis methods, several sufficient conditions ensuring the existence of solitary wave and periodic wave solutions are obtained. Some corresponding results in the literature are improved and extended. An example is given to illustrate the effectiveness of our results.

Solitary wave solutions for singular non-Newtonian filtration equations

This paper is mainly concerned with the solitary wave solutions for the singular non-Newtonian filtration equations. A sufficient criterion for the existence of solitary wave solutions of the considered equation is provided via an innovative method of Lyusternik-Schnirelmann category theory and variational methods. Some recent results in the literature have been complemented and improved.

Positive periodic wave solutions of singular non-Newtonian filtration equations

We study the existence and multiplicity of positive periodic wave solutions for one-dimensional non-Newtonian filtration equations with singular nonlinear sources. We discuss both the attractive singular case and the repulsive singular case. The proof is based on an extension of the continuation theorem of coincidence degree theory.

Critical Fujita exponents to a class of non-Newtonian filtration equations with fast diffusion

We consider the Cauchy problem to a class of fast-diffusion non-Newtonian filtration equations. Besides the usual degeneracy in the fast-diffusion non-Newtonian filtration, the equation is degenerate or singular at infinity, depending on the sign of the parameter related to the coefficient of diffusion. Fujita type theorems are established and the critical Fujita exponent is determined. Specially, we also prove that the nontrivial solution blows up in a finite time on the critical situation.

Quenching phenomenon for a non-Newtonian filtration equation with singular boundary flux

This paper is concerned with the quenching phenomenon for the one-dimensional non-Newtonian filtration equation with both source term and Neumann boundary condition. With two different kinds of initial data, we prove that the solution must quench in a finite time and the time derivative blows up at a quenching point. The corresponding quenching rate and a lower bound for the quenching time are also obtained.

Solitary Wave and Periodic Wave Solutions for a Non-Newtonian Filtration Equation

This paper is concerned with a non-Newtonian filtration equation with nonlinear sources. By using an extension of Mawhin’s continuation theorem and some analysis methods, we obtain some existence results of solitary wave and periodic wave solutions for the non-Newtonian filtration equation with nonlinear sources.

Study of Solutions for a Non-Newtonian Filtration Equation with Nonlocal Boundary Condition

In this paper, a non-Newtonian filtration equation with nonlocal boundary condition and a localized source is studied. The results to existence and blowup of solutions are given.

Fujita type theorems for a class of nonlinear diffusion equations

This paper studies a class of weighted non-Newtonian filtration equations with slow diffusion. By using the method introduced by Galaktionov and Levine for the classical non-Newtonian filtration equation, we establish the blow-up theorems of Fujiita type for the extended model, where more difficult and complicated estimates are required to treat the additional degeneracy and singularity. In particular, we prove via a delicate analysis that the critical situation of $p=p_c$ belongs to the blow-up case. The conclusions of this paper quantitatively show the influence of the degeneracy and singularity to the critical Fujita exponents of non-Newtonian filtration equations.

Asymptotic Behavior of Solutions to Fast Diffusive Non-Newtonian Filtration Equations Coupled by Nonlinear Boundary Sources

This paper is concerning the asymptotic behavior of solutions to the fast diffusive non-Newtonian filtration equations coupled by the nonlinear boundary sources. We are interested in the critical global existence curve and the critical Fujita curve, which are used to describe the large-time behavior of solutions. It is shown that the above two critical curves are both the same for the multidimensional problem we considered.

Traveling wavefronts for a time delayed non-Newtonian filtration equation

This paper is concerned with traveling wavefronts for the evolutionary p-Laplacian with time delay. We first establish some necessary and sufficient conditions for the existence of monotone nondecreasing and nonincreasing traveling wavefronts, which correspond to definite wave speeds c1∗ and c2∗ respectively. Moreover, special attention is also paid to the asymptotic behavior of traveling fronts connecting two uniform steady states.

On existence and nonexistence of the positive solutions of non-newtonian filtration equation

The subject of this investigation is existence and nonexistence of positive solutions of the following nonhomogeneous equation <p align="center"> $ \rho (|x|) \frac{\partial u}{\partial t}- \sum_{i=1}^N D_i(u^{m-1}|D_i u|^{\lambda -1}D_i u)+g(u)+lu=f(x)$ (1) <p align="left" class="times"> or, after the change $v=u^{\sigma}$, $\sigma =\frac{m+\lambda -1}{\lambda }, $ of equation <p align="center"> $\rho (|x|) \frac{\partial v^{\frac{1}{ \sigma }}}{\partial t}-\sigma ^{-\lambda }\sum_{i=1} ^N D_i(|D_i v|^{\lambda -1}D_i v)+g(v^{\frac{1}{\sigma }}) +lv^{\frac{1}{ \sigma }}=f(x),$ (1') <p align="left" class="times"> in unbounded domain $R_+\times R^N,$ where the term $g(s)$ is supposed to satisfy just a lower polynomial growth condition and $g'(s) > -l_1$. The existence of the solution in $ L^{1+1/\sigma}(0, T; L^{1+1/\sigma}(R^N))\cap L^{\lambda +1}(0, T; W^{1,\lambda +1}(R^N))$ is proved. Also, under some condition on $g(s)$ and $u_0$ is shown a nonexistence of the solution.