On global behavior of solutions to an inverse problem for nonlinear parabolic equations

Abstract

We find conditions on data guaranteeing global nonexistence of solutions to an inverse source problem for a class of nonlinear parabolic equations. We also establish a stability result on a bounded domain for a problem with the opposite sign on the power type nonlinearity.  2005 Elsevier Inc. All rights reserved.

Finite time blow up of solutions to an inverse problem for a quasilinear parabolic equation with power nonlinearity

“…Existence and uniqueness of solutions to inverse problems for parabolic equations are studied by several authors [4,5,7,8,9]. Asymptotic stability of solutions to such problems are investigated in [2,9,10,11].…”

“…But less is known about inverse problem for nonlinear parabolic equations. Eden and Kalantarov [2] studied the following problem…”

Global nonexistence and stability of solutions of inverse problems for a class of Petrovsky systems

“…But less is known about the global nonexistence for solutions of hyperbolic and parabolic inverse problems. The interested reader is referred to the papers [4,5].…”

“…In [5], Eden and Kalantarov applied the modified concavity method to the problem u t u juj p u C b.x; t; u; ru/ D F .t/!.x/; x 2 ; t > 0;…”

“…In this work, by modifying the methods in [5], we study the global in time behavior of solutions to an inverse problem for a class of Petrovsky systems. Moreover, we establish an asymptotic stability result for the corresponding problem with the opposite sign of power-type nonlinearities.…”