Bounds for lifespan of solutions to strongly damped semilinear wave equations with logarithmic sources and arbitrary initial energy

Abstract

The main aim of this paper is to deal with the upper and lower bounds for blow-up time of solutions to the following equation:which has been studied in [5]. For high initial energy, it is well known that the classical potential well method is not effective. In order to overcome this difficulty, the authors apply the new energy estimate method to establish the lower bound of the L 2 (Ω) norm of the solution. Furthermore, the authors construct a new control functional and combine energy inequalities with the concavity argument to prove that the solution blows up in finite time for high initial energy. Meanwhile, an estimate of the upper bound of blow-up time is also obtained. Finally, a lower bound for blow-up time is obtained by introducing a new control functional. These results fill the gap of [5].

Initial boundary value problem for a strongly damped wave equation with a general nonlinearity

“…By comparing the results of [3,14] with that of [2,16], we find that power type nonlinearity and logarithmic nonlinearity have similar effect on finite time blow-up of solutions to initial-boundary value problems for damped semilinear wave equations, which is the main purpose of this paper. More precisely, we shall consider the blow-up property of solutions to Problem (1) with a general nonlinearity.…”

“…The main reason for this restriction comes from the Sobolev embedding H 1 0 (Ω) → L 2p−2 (Ω), which does not hold for the supercritical exponent p ∈ ( 2n−2 n−2 , 2n n−2 ). Then, the second problem is whether a lower bound for the blowup time can be obtained for supercritical exponent p. Recently, Zu and Guo [16] obtained a new blow-up criterion for Problem (4) which contains the case that the initial energy is supercritical. Moreover, when p ∈ ( 2n−2 n−2 , 2n n−2 ), they also derived a lower bound for blow-up time with the help of a first order differential inequality for a newly constructed auxiliary functional.…”

“…In this way, it is no more necessary to distinguish the cases f (u) = |u| p−2 u and f (u) = |u| p−2 u ln |u| for Problem (1) in future proofs, and the methods used here are applicable to both (3) and (4). Moreover, the sufficient conditions for finite time blow-up of solutions to Problem (1) are weaker than those in [14] and [16].…”

“…Remark 3. Unlike the blow-up conditions given in [14] and [16] which require that the initial data satiefy (u 0 , u 1 ) > CE(0) for some positive constant C, our blow-up condition for Problem (1) is of the form u 0 2 + 2(u 0 , u 1 ) > CE(0). This makes it more flexible to choose such initial data.…”

The lifespan of solutions for a viscoelastic wave equation with a strong damping and logarithmic nonlinearity

“…They derived the finite time blow-up results of weak solutions, and presented the lower and upper bounds for blow-up time by combining the concavity method, perturbation energy method with differential-integral inequality technique. Subsequently, Zu and Guo [19] improved their results by discussing the finite time blow-up for high initial energy and by giving an estimate of the lower bound for blow-up time when 1 + n n−2 < p < 2n n−2 . Until recently, Ha and Park [8] investigated the viscoelastic wave equation with a strong damping and logarithmic nonlinearity, i.e.…”