Blow-Up Time of Some Nonlinear Wave Equations

Abstract

Abstract. In this paper, we consider the following initial-boundary value problem is a positive, increasing and convex function for nonnegative values of s. Under some assumptions, we show that, if ε is small enough, then the solution u of the above problem blows up in a finite time, and its blow-up time tends to that of the solution of the following differential equationFinally, we give some numerical results to illustrate our analysis.

Global existence and quenching for a damped hyperbolic MEMS equation with the fringing field

“…The estimates in our proof are quite similar to those in [24,32]. Differentiating z(t ) twice, we obtain z t t + z t = −µz + λ…”

“…For an elliptic and parabolic operator A, thanks to the maximum principle, we have the results of the time-global existence [12,19,23,25] for sufficiently small λ > 0, the quenching [12,18,23,25] for sufficiently large λ > 0, the connecting orbit [23], the Morse-Smale property [23], the location of the quenching point [17] and its stationary solution [5,6,7,11,13,23]. Also in the hyperbolic problem, we have similar results to those in the parabolic case, i.e., the global existence [3,26,38], the quenching [3,26,32,38], the estimate of the quenching time [32] and the singularity of the derivative [2]. In the damped hyperbolic case, we have the global existence [27] and quenching [15,27].…”

Existences and Boundary Behavior of Boundary Blow-Up Solutions to Quasilinear Elliptic Systems With Singular Weights

“…Very recently, Zhang [33] and Yang [23] extended the above results to the problem (1.3) and gained some new results with nonlinear gradient terms. Problem (1.3) was discussed in a number of works; see, [2,3,4,5,9,10,11,12,13,19,23,25,34], Now let us return to problem (1.1). When m = n = 2, system (1.1) becomes 4) in the paper [14], when a(x) = 1, b(x) = 1, under Dirichlet boundary conditions of three different types: both components of (u, v) are bounded on ∂Ω (finite case); one of them is bounded while the other blows up(semilinear case); or both components blow up simultaneously(infinite case), under the assumption that(a − 1)(e − 1) > bc, necessary and suffcient conditions for existence of positive solutions were found, and uniqueness or multiplicity were also obtained, together with the exact boundary behavior of solutions.…”