scholarly journals Global existence and uniqueness of the solution to a nonlinear parabolic equation

2018 ◽  
Vol 14 (2) ◽  
pp. 7860-7863
Author(s):  
Alexander G. Ramm
Keyword(s):  
Parabolic Equation ◽  
Global Existence ◽  
Global Solution ◽  
Existence And Uniqueness ◽  
Nonlinear Parabolic Equation ◽  
Unique Global Solution ◽  
Nonlinear Parabolic ◽  
Uniqueness Of The Solution ◽  
Global Existence And Uniqueness

Consider the equation  u’ (t)  -  u + | u |p u = 0, u(0) = u0(x), (1), where u’ := du/dt , p = const > 0, x E R3, t > 0.  Assume that u0 is a smooth and decaying function,           ||u0|| =            sup             |u(x, t)|.                                         x E R3 ,t E R+      It is proved that problem (1) has a unique global solution and this solution satisfies the following estimate                              ||u(x, t)|| < c, where c > 0 does not depend on x, t.

scholarly journals Asymptotic behavior of solutions for a semibounded nonmonotone evolution equation

2003 ◽  
Vol 2003 (9) ◽  
pp. 521-538
Author(s):  
Nikos Karachalios ◽  
Nikos Stavrakakis ◽  
Pavlos Xanthopoulos
Keyword(s):  
Dynamical System ◽  
Asymptotic Behavior ◽  
Parabolic Equation ◽  
Evolution Equation ◽  
Existence And Uniqueness ◽  
Nonlinear Parabolic Equation ◽  
Evolution Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Uniqueness Of Solutions ◽  
Nonlinear Parabolic

We consider a nonlinear parabolic equation involving nonmonotone diffusion. Existence and uniqueness of solutions are obtained, employing methods for semibounded evolution equations. Also shown is the existence of a global attractor for the corresponding dynamical system.

Global Existence for Nonlinear Parabolic Problems With Measure Data– Applications to Non-uniqueness for Parabolic Problems With Critical Gradient terms

2011 ◽  
Vol 11 (4) ◽  
Author(s):  
Boumediene Abdellaoui ◽  
Andrea Dall’Aglio ◽  
Ireneo Peral ◽  
Sergio Segura de Léon
Keyword(s):  
Parabolic Equation ◽  
Global Existence ◽  
Present Article ◽  
Open Subset ◽  
Radon Measure ◽  
Nonlinear Parabolic Equation ◽  
Parabolic Problems ◽  
Measure Data ◽  
Nonlinear Parabolic ◽  
Bounded Open Subset

AbstractIn the present article we study global existence for a nonlinear parabolic equation having a reaction term and a Radon measure datum:where 1 < p < N, Ω is a bounded open subset of ℝ

On a quasilinear wave equation with nonlinear damping

1988 ◽  
Vol 110 (3-4) ◽  
pp. 227-239 ◽  
Author(s):  
Dang Dinh Hai
Keyword(s):  
Wave Equation ◽  
Initial Data ◽  
Global Existence ◽  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Nonlinear Damping ◽  
Quasilinear Wave Equation ◽  
Uniqueness Of The Solution ◽  
Global Existence And Uniqueness ◽  
Image Position

SynopsisWe prove the global existence and uniqueness of the solution of the initial and boundary value problem for the equationby using the classical Galerkin method when the forcing term and the initial data are in some sense small. The asymptotic behaviour of the solution as t → ∞ is also considered.

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