scholarly journals Solutions to Nonlinear Evolutionary Parabolic Equations of the Diffusion Wave Type

Symmetry ◽  
2021 ◽  
Vol 13 (5) ◽  
pp. 871
Author(s):  
Alexander Kazakov
Keyword(s):  
Parabolic Equations ◽  
Unknown Function ◽  
Heat Waves ◽  
Parabolic Type ◽  
Porous Medium Equation ◽  
Heat Propagation ◽  
Second Order ◽  
Cauchy Problems ◽  
Medium Equation ◽  
Special Cases

The article deals with nonlinear second-order evolutionary partial differential equations (PDEs) of the parabolic type with a reasonably general form. We consider the case of PDE degeneration when the unknown function vanishes. Similar equations in various forms arise in continuum mechanics to describe some diffusion and filtration processes as well as to model heat propagation in the case when the properties of the process depend significantly on the unknown function (concentration, temperature, etc.). One of the exciting and meaningful classes of solutions to these equations is diffusion (heat) waves, which describe the propagation of perturbations over a stationary (zero) background with a finite velocity. It is known that such effects are atypical for parabolic equations; they arise as a consequence of the degeneration mentioned above. We prove the existence theorem of piecewise analytical solutions of the considered type and construct exact solutions (ansatz). Their search reduces to the integration of Cauchy problems for second-order ODEs with a singularity in the term multiplying the highest derivative. In some special cases, the construction is brought to explicit formulas that allow us to study the properties of solutions. The case of the generalized porous medium equation turns out to be especially interesting as the constructed solution has the form of a soliton moving at a constant velocity.

scholarly journals Diffusion-Wave Type Solutions to the Second-Order Evolutionary Equation with Power Nonlinearities

Mathematics ◽  
2022 ◽  
Vol 10 (2) ◽  
pp. 232
Author(s):  
Alexander Kazakov ◽  
Anna Lempert
Keyword(s):  
Porous Medium Equation ◽  
Second Order ◽  
Phase Portraits ◽  
Evolutionary Equation ◽  
Finite Difference Schemes ◽  
Cauchy Problems ◽  
Wave Type ◽  
Diffusion Wave ◽  
Medium Equation ◽  
Special Cases

The paper deals with a nonlinear second-order one-dimensional evolutionary equation related to applications and describes various diffusion, filtration, convection, and other processes. The particular cases of this equation are the well-known porous medium equation and its generalizations. We construct solutions that describe perturbations propagating over a zero background with a finite velocity. Such effects are known to be atypical for parabolic equations and appear as a consequence of the degeneration of the equation at the points where the desired function vanishes. Previously, we have constructed it, but here the case of power nonlinearity is considered. It allows for conducting a more detailed analysis. We prove a new theorem for the existence of solutions of this type in the class of piecewise analytical functions, which generalizes and specifies the earlier statements. We find and study exact solutions having the diffusion wave type, the construction of which is reduced to the second-order Cauchy problem for an ordinary differential equation (ODE) that inherits singularities from the original formulation. Statements that ensure the existence of global continuously differentiable solutions are proved for the Cauchy problems. The properties of the constructed solutions are studied by the methods of the qualitative theory of differential equations. Phase portraits are obtained, and quantitative estimates are determined by constructing and analyzing finite difference schemes. The most significant result is that we have shown that all the special cases for incomplete equations take place for the complete equation, and other configurations of diffusion waves do not arise.

scholarly journals On the Solutions of a Porous Medium Equation with Exponent Variable

2019 ◽  
Vol 2019 ◽  
pp. 1-15 ◽  
Author(s):  
Huashui Zhan
Keyword(s):  
Porous Medium ◽  
Weak Solutions ◽  
Boundary Value ◽  
Porous Medium Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Medium Equation ◽  
Special Cases ◽  
The Stability ◽  
Initial Boundary Value

The paper studies the initial-boundary value problem of a porous medium equation with exponent variable. How to deal with nonlinear term with the exponent variable is the main dedication of this paper. The existence of the weak solution is proved by the monotone convergent method. Moreover, according to the different boundary value conditions, the stability of weak solutions is studied. In some special cases, the stability of weak solutions can be proved without any boundary value condition.

LAGRANGIAN FEYNMAN FORMULAS FOR SECOND-ORDER PARABOLIC EQUATIONS IN BOUNDED AND UNBOUNDED DOMAINS

2010 ◽  
Vol 13 (03) ◽  
pp. 377-392 ◽  
Author(s):  
YANA A. BUTKO ◽  
MARTIN GROTHAUS ◽  
OLEG G. SMOLYANOV
Keyword(s):  
Parabolic Equations ◽  
Diffusion Processes ◽  
Unbounded Domains ◽  
Variable Coefficients ◽  
Second Order ◽  
Cauchy Problems ◽  
Elementary Functions ◽  
Functional Integrals ◽  
Finite Dimensional ◽  
Variable Diffusion

In this note a class of second-order parabolic equations with variable coefficients, depending on coordinate, is considered in bounded and unbounded domains. Solutions of the Cauchy–Dirichlet and the Cauchy problems are represented in the form of a limit of finite-dimensional integrals of elementary functions (such representations are called Feynman formulas). Finite-dimensional integrals in the Feynman formulas give approximations for functional integrals in the corresponding Feynman–Kac formulas, representing solutions of these problems. Hence, these Feynman formulas give an effective tool to calculate functional integrals with respect to probability measures generated by diffusion processes with a variable diffusion coefficient and absorption on the boundary.

Symmetrization and Mass Comparison for Degenerate Nonlinear Parabolic and Related Elliptic Equations

2005 ◽  
Vol 5 (1) ◽  
Author(s):  
Juan Luis Vázquez
Keyword(s):  
Parabolic Equations ◽  
A Priori ◽  
Porous Medium Equation ◽  
Nonlinear Parabolic Equations ◽  
Detailed Knowledge ◽  
Medium Equation ◽  
Mass Comparison ◽  
Nonlinear Parabolic ◽  
Comparison Results ◽  
Special Solutions

AbstractWe consider the solutions to various nonlinear parabolic equations and their elliptic counterparts and prove comparison results based on two main tools, symmetrization and mass concentration comparison. The work focuses on equations like the porous medium equation, the filtration equation and the p-Laplacian equation. The results will be used in a companion work in combination with a detailed knowledge of special solutions to obtain sharp a priori bounds and decay estimates for wide classes of solutions of those equations.

LYAPUNOV FUNCTIONALS IN SINGULAR LIMITS FOR PERTURBED QUASILINEAR DEGENERATE PARABOLIC EQUATIONS

2003 ◽  
Vol 01 (04) ◽  
pp. 351-385 ◽  
Author(s):  
MANUELA CHAVES ◽  
VICTOR A. GALAKTIONOV
Keyword(s):  
Porous Medium ◽  
Parabolic Equations ◽  
Quasilinear Parabolic Equation ◽  
Porous Medium Equation ◽  
Degenerate Parabolic Equations ◽  
Radial Solutions ◽  
Medium Equation ◽  
Singular Limits ◽  
Self Similar ◽  
Degenerate Parabolic

As a key example, we study the asymptotic behaviour near finite focusing time t=T of radial solutions of the porous medium equation with absorption [Formula: see text] with bounded compactly supported initial data u(x,0)=u0(|x|), and exponents m>1 and p>pc, where pc=pc(m,N)∈(-m,0) is a critical exponent. We show that under certain assumptions, the behaviour of the solution as t→T- near the origin is described by self-similar Graveleau solutions of the porous medium equation ut=Δum. In the rescaled variables, we deal with an exponential non-autonomous perturbation of a quasilinear parabolic equation, which is shown to admit an approximate Lyapunov functional. The result is optimal, and in the critical case p=pc an extra ln (T-t) scaling of the Graveleau asymptotics is shown to occur. Other types of self-similar and non self-similar focusing patterns are discussed.

scholarly journals The Stability of the Solutions for a Porous Medium Equation with a Convection Term

2018 ◽  
Vol 2018 ◽  
pp. 1-11 ◽  
Author(s):  
Huashui Zhan ◽  
Miao Ouyang
Keyword(s):  
Porous Medium ◽  
Boundary Value ◽  
Porous Medium Equation ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Convection Term ◽  
Medium Equation ◽  
Special Cases ◽  
The Stability ◽  
Initial Boundary Value

This paper studies the initial-boundary value problem of a porous medium equation with a convection term. If the equation is degenerate on the boundary, then only a partial boundary condition is needed generally. The existence of the weak solution is proved by the monotone convergent method. Moreover, according to the different boundary value conditions, the stability of the solutions is studied. In some special cases, the stability can be proved without any boundary value condition.

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