Method of Upper and Lower Solutions for Parabolic Equations

2022 ◽  
pp. 283-290
Keyword(s):  
Parabolic Equations ◽  
Upper And Lower Solutions

scholarly journals Existence of extremal periodic solutions for quasilinear parabolic equations

1997 ◽  
Vol 2 (3-4) ◽  
pp. 257-270 ◽  
Author(s):  
Siegfried Carl
Keyword(s):  
Parabolic Equations ◽  
Evolution Equations ◽  
Quasilinear Parabolic Equation ◽  
Upper And Lower Solutions ◽  
Nonlinear Evolution Equations ◽  
Dirichlet Boundary Conditions ◽  
Quasilinear Parabolic Equations ◽  
All Solutions ◽  
Truncation Techniques ◽  
Quasilinear Parabolic

In this paper we consider a quasilinear parabolic equation in a bounded domain under periodic Dirichlet boundary conditions. Our main goal is to prove the existence of extremal solutions among all solutions lying in a sector formed by appropriately defined upper and lower solutions. The main tools used in the proof of our result are recently obtained abstract results on nonlinear evolution equations, comparison and truncation techniques and suitably constructed special testfunction.

Existence of bounded solutions of some nonlinear parabolic equations

1987 ◽  
Vol 107 (3-4) ◽  
pp. 313-326 ◽  
Author(s):  
A. Mokrane
Keyword(s):  
Parabolic Equations ◽  
Upper And Lower Solutions ◽  
Nonlinear Parabolic Equations ◽  
Parabolic Problems ◽  
Bounded Solutions ◽  
Open Set ◽  
Nonlinear Parabolic ◽  
Carathéodory Functions ◽  
Characteristic Features ◽  
Image Position

SynopsisThis paper proves the existence of (at least) one solution of the following equation:Here, is an elliptic operator of Leray-Lions type acting from into Lp′(0, T; W−1.p′ (Ω)), (1/p + 1/p′ = 1) and |F(u, ∇u)| ≧b(|u|)(l + |≧u|P). There are no smoothness assumptions on the bounded open set Ω; the operator and the nonlinearity F(u, ∇u) are denned in terms of Carathéodory functions. These points are the most characteristic features of this paper.Assuming the existence of upper and lower solutions allows us to obtain L∞(Q)-estimates. An estimate is then proved. The final step is to prove the strong convergence in of the approximations. This proof relies on the method introduced by L. Boccardo, F. Murat and J. P. Puel for elliptic and parabolic problems of this type.

The form of blow-up for nonlinear parabolic equations

1984 ◽  
Vol 98 (1-2) ◽  
pp. 183-202 ◽  
Author(s):  
A. A. Lacey
Keyword(s):  
Parabolic Equations ◽  
Blow Up ◽  
Upper And Lower Solutions ◽  
Nonlinear Parabolic Equations ◽  
One Dimensional ◽  
Nonlinear Parabolic ◽  
Infinite Region ◽  
Limiting Behaviour ◽  
Semilinear Parabolic ◽  
Modified Equation

SynopsisSemilinear parabolic equations of the form u1 = ∇2u + δf(u), where f is positive and is finite, are known to exhibit the phenomenon of blow-up, i.e. for sufficiently large S, u becomes infinite after a finite time t*. We consider one-dimensional problems in the semi-infinite region x>0 and find the time to blow-up (t*). Also, the limiting behaviour of u as t→t*- and x→∞ is determined; in particular, it is seen that u blows up at infinity, i.e. for any given finite x, u is bounded as t→t*. The results are extended to problems with convection.The modified equation xu, = uxx +f(u) is discussed. This shows the possibility of blow-up at x =0 even if u(0, f) = 0. The manner of blow-up is estimated.Finally, bounds on the time to blow-up for problems in finite regions are obtained by comparing u with upper and lower solutions.

Hypothesis on the Solvability of Parabolic Equations with nonlocal Condition

2002 ◽  
Vol 7 (1) ◽  
pp. 93-104 ◽  
Author(s):  
Mifodijus Sapagovas
Keyword(s):  
Parabolic Equation ◽  
Parabolic Equations ◽  
Existence And Uniqueness ◽  
Nonlocal Condition ◽  
Nonlocal Conditions ◽  
Numerical Algorithms ◽  
Uniqueness Of The Solution

Numerous and different nonlocal conditions for the solvability of parabolic equations were researched in many articles and reports. The article presented analyzes such conditions imposed, and observes that the existence and uniqueness of the solution of parabolic equation is related mainly to ”smallness” of functions, involved in nonlocal conditions. As a consequence the hypothesis has been made, stating the assumptions on functions in nonlocal conditions are related to numerical algorithms of solving parabolic equations, and not to the parabolic equation itself.

Sign in / Sign up

Export Citation Format

Share Document