Global behavior of solutions to an inverse problem for semilinear hyperbolic equations

This paper is concerned with global in time behavior of solutions for a semilinear, hyperbolic, inverse source problem. We prove two types of results. The first one is a global nonexistence result for smooth solutions when the data is chosen appropriately. The second type of results is the asymptotic stability of solutions when the integral constraint vanishes as t goes to infinity. Bibliography: 22 titles. Dedicated to the memory of Olga Aleksandrovna Ladyzhenskaya Published in Zapiski Nauchnykh Seminarov POMI, Vol. 318, 2004, pp. 120–134.     

Global small solutions of the cauchy problem for a semilinear system of equations of thermoelasticity

For a semilinear system of equations of thermoelasticity, we establish a theorem on the existence and uniqueness of global solutions in a multidimensional space under the condition that the initial data are sufficiently small. We also obtain estimates for the decrease of solutions as time increases.     

Existence and nonexistence of global solutions of the cauchy problem for semilinear hyperbolic equations with dissipation and with anisotropic elliptic part

We consider the Cauchy problem for a semilinear hyperbolic equation with anisotropic elliptic part and with dissipation. We prove existence and nonexistence theorems for global solutions.     

On the cauchy problem for certain system of semilinear parabolic equations

In this paper, we use the abstract theory of semilinear parabolic equations and a priori estimate techniques to prove the global existence and uniqueness of smooth solutions to the Cauchy problem for the following system of parabolic equations:

Global solvability for abstract semilinear evolution equations

The Cauchy problem for the abstract semilinear evolution equation u^′(t) = Au (t) + B (u (t)) + C (u (t)) is discussed in a general Banach space X. Here A is the so‐called Hille‐Yosida operator in X, B is a differentiable operator from D (A) into X, and C is a locally Lipschitz continuous operator from D (A) into itself. A vectorvalued functional defined only on X is used and appropriate conditions on the nonlinear operators B and C are imposed so that a vector‐valued functional defined on the domain of the operator A may be constructed in order to specify the growth of a global solution. The advantage of our formulation lies in the fact that it is possible to obtain a global solution by checking some energy inequalities concerning only low order derivatives (© 2010 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Existence of the global solutions of the Cauchy problem for a system of semilinear pseudohyperbolic equations with structural dissipation

Mathematical Notes -     

Global solutions of a two-dimensional initial boundary-value problem for a system of semilinear magnetoelasticity equations

We prove the theorem on the existence and uniqueness of global solutions of a system of semilinear magnetoelasticity equations in a two-dimensional space.     

On the cauchy problem for a family of quasilinear hyperbolic equations

We prove local well-posedness and give some global existence and blow-up criteria for solutions of a family of quasilinear hyperbolic equations arising in shallow water theory.     

Global well-posedness for semilinear hyperbolic equations with dissipative term

We study the initial boundary value problem of semilinear hyperbolic equations with dissipative term. By introducing a family of potential wells we derive the invariant sets and vacuum isolating of solutions. Then we prove the global existence, nonexistence and asymptotic behaviour of solutions. In particular we obtain some sharp conditions for global existence and nonexistence of solutions.     

Global weak solutions of the Cauchy problem for semilinear pseudo-hyperbolic equations

The Cauchy problem for a class of semilinear pseudo-hyperbolic equations is considered. For the corresponding linear problems, we obtain L _p → L _q estimates. By using these estimates, we prove global solvability theorems. We also establish the behavior of solutions as t → + ∞.