Uniqueness and stability of the generalized solution of the Cauchy problem for a quasi-linear equation

1963 ◽  
pp. 285-290 ◽  
Author(s):  
O. A. Oleĭnik
Keyword(s):  
Cauchy Problem ◽  
Linear Equation ◽  
Generalized Solution ◽  
The Cauchy Problem

GENERALIZED SOLUTIONS OF THE CAUCHY PROBLEM FOR SYSTEMS OF NONLINEAR PARABOLIC EQUATIONS AND DIFFUSION PROCESSES

2012 ◽  
Vol 12 (01) ◽  
pp. 1150001 ◽  
Author(s):  
YANA BELOPOLSKAYA ◽  
WOJBOR A. WOYCZYNSKI
Keyword(s):  
Cauchy Problem ◽  
Parabolic Equations ◽  
Diffusion Processes ◽  
Stochastic Approach ◽  
Generalized Solution ◽  
Nonlinear Parabolic Equations ◽  
Kolmogorov Equation ◽  
Generalized Solutions ◽  
Nonlinear Parabolic ◽  
The Cauchy Problem

The purpose of this paper is to construct both strong and weak solutions (in certain functional classes) of the Cauchy problem for a class of systems of nonlinear parabolic equations via a unified stochastic approach. To this end we give a stochastic interpretation of such a system, treating it as a version of the backward Kolmogorov equation for a two-component Markov process with coefficients depending on the distribution of its first component. To extend this approach and apply it to the construction of a generalized solution of a system of nonlinear parabolic equations, we use results from Kunita's theory of stochastic flows.

scholarly journals General decay and well-posedness of the Cauchy problem for the Jordan-Moore-Gibson-Thompson equation with memory

Filomat ◽  
2021 ◽  
Vol 35 (5) ◽  
pp. 1745-1773
Author(s):  
Salah Boulaaras ◽  
Abdelbaki Choucha ◽  
Djamel Ouchenane
Keyword(s):  
Cauchy Problem ◽  
Existence Result ◽  
Local Existence ◽  
Generalized Solution ◽  
General Decay ◽  
Small Data ◽  
Well Posedness ◽  
Third Order ◽  
The Cauchy Problem ◽  
With Memory

In this paper, we consider the Cauchy problem of a third order in time nonlinear equation known as the Jordan-Moore-Gibson-Thompson (JMGT) equation with the presence of both memory. Using the well known energy method combined with Lyapunov functionals approach, we prove a general decay result, and we show a local existence result in appropriate function spaces. Finally, we prove a global existence result for small data, and we prove the uniqueness of the generalized solution.

Sign in / Sign up

Export Citation Format

Share Document