Initial boundary value problem for semilinear hyperbolic equations and parabolic equations with critical initial data

Abstract

Abstract. We study the initial boundary value problem of semilinear hyperbolic equations u tt − Δu = f (u) and semilinear parabolic equations

Asymptotic behavior and blow up of solutions for semilinear parabolic equations at critical energy level

“…J(u 0 ) ≥ M (M is an arbitrary positive constant) can lead to both vanishing and blow-up without any further exact descriptions of the sets of the corresponding initial data for global existence or blow up. And we do not know how the solutions act with time if the initial data of problem [5]. And by employing the invariant manifold for J(u) < d for the parabolic case the same blowup result was obtained.…”

Invariant sets and the blow up threshold for coupled systems of reaction–diffusion

“…For initial datum u 0 belongs to Z , the associated solution blows up in a finite time. Yacheng and Junsheng [3,4], Runzhang [5] considered the same initial-boundary value problem with critical energy initial data and improved Payne and Sattinger's result. [2]Aspecial case of (2), equation with f (u) = |u| p−1 u has been extensively studied for both sub-critical exponent case (1 < p < n+2 n−2 ) and critical exponent case ( p = n+2 n−2 ).…”

“…For initial datum u 0 belongs to the unstable set, the associated solution blows up in a finite time. We refer to [2,5,[8][9][10][11][12][13] and papers cited therein. Recently, a new method to find stable and unstable set is established by Ma [14,15].…”

Global results for semilinear hyperbolic equations with damping term on manifolds with conical singularity

“…The nonexistence of global solutions of semilinear hyperbolic equations with no damping terms in the boundary conditions is investigated by J.L. Lions , X. Runzhang , R.T. Glassey , and H.A. Levine .…”

“…It is well known that problem has been studied by many authors, for example, . Then, X. Runzhang extended the results corresponding to problem in and to the critical case, and the authors in studied the case with damping term and nonlinear term kinds of problem . In this paper, we shall find the existence and nonexistence theorems for problem in cone Sobolev space , which will be given in the next section.…”