Long-term analysis of strongly damped nonlinear wave equations

Abstract

We consider the strongly damped nonlinear wave equation u tt − u t − u + f (u t ) + g(u) = h with Dirichlet boundary conditions, which serves as a model in the description of thermal evolution within the theory of type III heat conduction. In particular, the nonlinearity f acting on u t is allowed to be nonmonotone and to exhibit a critical growth of polynomial order 5. The main focus is the long-term analysis of the related solution semigroup, which is shown to possess the global attractor in the natural weak energy space.

Structural stability analysis of solutions to the initial boundary value problem for a nonlinear strongly damped wave equation

“…Different conclusions were obtained in many other articles [2,7,10,13,15,16]. References [1,[3][4][5]12] can be given for more information on the structural stability result for interested readers.…”

Sufficient condition for the existence of an inertial manifold for a hyperbolic equation with weak and strong dissipation

“…Under these assumptions, problem (1), (2) has a unique solution u ∈ C [0, T ]; H 1 0 (Ω) , ∂ t u ∈ C [0, T ]; L 2 (Ω) for any T > 0 (see [4][5][6]). Hence, this problem generates a continuous semigroup {S(t)}, t 0, acting on the phase space H = H 1 0 (Ω) × L 2 (Ω) by the formula S(t)(u 0 (x), p 0 (x)) = y(t) ≡ (u(t, x), p(t, x)) ∈ H, This work was financially supported by the Russian Foundation for Basic Research (grant no.…”

Regularity of Global Attractors for the Kirchhoff Wave Equation

“…In this paper, we discuss the regularity of global attractors for the following Kirchhoff wave equation where Ω is a bounded domain in which described the thermal evolution and ( ) h u denoted a source term depending nonlinearly on displacement, ( ) t f u denoted a nonlinearly temperature-dependent internal source term [3]. With different conditions about the growth exponents q and p of the nonlinearities ( ) t f u and ( ) h u , some scholars [4] [5] analyzed the longtime behaviour of solutions of (1.3)-(1.2) by the global and exponential attractors in a bounded region of 3  .…”