Semilinear Damped Wave Equation Research Articles

Compact difference scheme for the two-dimensional semilinear wave equation

In this article, we investigate the use of a high-order compact finite difference method for solving a two-dimensional damped semilinear wave equation. We use a new iterative method that employs compact finite difference operators to approximate the second-order spatial derivatives and achieve fourth-order convergence. We establish the stability and convergence of the compact finite difference scheme in the sense of the discrete H1-norm. Finally, we present numerical results to demonstrate the accuracy and efficiency of the proposed scheme.

Exponential convergence to steady-states for trajectories of a damped dynamical system modeling adhesive strings

We study the global well-posedness and asymptotic behavior for a semilinear damped wave equation with Neumann boundary conditions, modeling a one-dimensional linearly elastic body interacting with a rigid substrate through an adhesive material. The key feature of of the problem is that the interplay between the nonlinear force and the boundary conditions allows for a continuous set of equilibrium points. We prove an exponential rate of convergence for the solution towards a (uniquely determined) equilibrium point.

Global existence and asymptotic behavior for semilinear damped wave equations on measure spaces

Global existence and asymptotic behavior for semilinear damped wave equations on measure spaces

The small mass limit for long time statistics of a stochastic nonlinear damped wave equation

We study the long time statistics of a class of semi–linear damped wave equations with polynomial nonlinearities and perturbed by additive Gaussian noise in dimensions 2 and 3. We find that if sufficiently many directions in the phase space are stochastically forced, the system is exponentially attractive toward its unique invariant measure with a convergent rate that is uniform with respect to the mass. Then, in the small mass limit, we prove the convergence of the first marginal of the invariant measures in a suitable Wasserstein distance toward the unique invariant measure of a stochastic reaction–diffusion equation. Together with the uniform geometric ergodicity, we obtain the validity of the small mass limit for the solutions on the infinite time horizon [0,∞). This extends previously known results established for the damped wave equations under Lipschitz nonlinearities.

Continuity of the attractors in time-dependent spaces and applications

In this paper, we investigate the continuity of the attractors in time-dependent phase spaces. (i) We establish two abstract criteria on the upper semicontinuity and the residual continuity of the pullback D-attractor with respect to the perturbations, and an equivalence criterion between their continuity and the pullback equi-attraction, which generalize the continuity theory of attractors developed recently in [27,28] to that in time-dependent spaces. (ii) We propose the notion of pullback D-exponential attractor, which includes the notion of time-dependent exponential attractor [33] as its spacial case, and establish its existence and Hölder continuity criterion via quasi-stability method introduced originally by Chueshov and Lasiecka [12,13]. (iii) We apply above-mentioned criteria to the semilinear damped wave equations with perturbed time-dependent speed of propagation: ϵρ(t)utt+αut−Δu+f(u)=g, with perturbation parameter ϵ∈(0,1], to realize above mentioned continuity of pullback D and D-exponential attractors in time-dependent phase spaces, and the method developed here allows to overcome the difficulty of the hyperbolicity of the model. These results deepen and extend recent theory of attractors in time-dependent spaces in literatures [15,20,29].

On the critical exponent and sharp lifespan estimates for semilinear damped wave equations with data from Sobolev spaces of negative order

We study semilinear damped wave equations with power nonlinearity |u|^p and initial data belonging to Sobolev spaces of negative order dot{H}^{-gamma }. In the present paper, we obtain a new critical exponent p=p_{textrm{crit}}(n,gamma ):=1+frac{4}{n+2gamma } for some gamma in (0,frac{n}{2}) and low dimensions in the framework of Sobolev spaces of negative order. Precisely, global (in time) existence of small data Sobolev solutions of lower regularity is proved for p>p_{textrm{crit}}(n,gamma ), and blow-up of weak solutions in finite time even for small data if 1<p<p_{textrm{crit}}(n,gamma ). Furthermore, in order to more accurately describe the blow-up time, we investigate sharp upper bound and lower bound estimates for the lifespan in the subcritical case.

A new class of small initial data which may shift the critical power and lifespan estimates for the classical damped wave equations

In this manuscript, lifespan of solutions to semilinear classical damped wave equations is investigated when the sum of initial position and speed is $ 0 $ pointwisely. The results in the existing literature are considered mainly in the case when the sum is positive. Here, it is shown that lifespan is extended with appropriate initial data with modified ODE approach. The result implies that the blowup mechanics for semilinear damped wave equations cannot be directly described by using the scaling property and the corresponding weak form approach. The results in this work leave different directions for further study of the lifespan behavior of the solutions to semilinear damped wave equations.

Blow-up of positive-initial-energy solutions for nonlinearly damped semilinear wave equations

Blow-up of positive-initial-energy solutions for nonlinearly damped semilinear wave equations

Initial boundary value problem for a damped wave equation with logarithmic nonlinearity

In this paper, a damped semilinear wave equation with logarithmic non-linearity is considered. Finite time blow-up criteria are established for solutions with both lower and higher initial energy, and an upper bound for the blow-up time is derived for each case. Moreover, by making full use of the strong damping term, a lower bound for the blow-up time is also obtained, for both subcritical and supercritical exponent. This partially extends some recent results obtained in [Initial boundary value problem for a class of strongly damped semilinear wave equations with logarithmic nonlinearity, Nonlinear Analysis, Real World Applications, 51(2020), 102968] by Di et al.

A remark on the blowing up of solutions to Nakao's problem

This paper is devoted to studying blow-up of solutions to some system of a semilinear damped wave equation and a semilinear wave equation for small data. In particular, we prove that if the exponents of the nonlinear terms satisfy suitable conditions, all solutions blow up in finite time even for small initial data, and we are also concerned with the upper estimate of lifespan of blowing-up solutions. A study on blowing up of solutions for this problem was originated by Wakasugi (2017), which used the test function method, and was subsequently partially improved by Chen-Reissig (2021) with an iteration argument. However, there is a gap between these two results. Our main aim in this paper is to bridge this gap while also providing a more concise proof.

Blow‐up and lifespan estimates for Nakao's type problem with nonlinearities of derivative type

In the present paper, we investigate blow‐up and lifespan estimates for a class of semilinear hyperbolic coupled system in with , which is a part of the so‐called Nakao's type problem weakly coupled a semilinear damped wave equation with a semilinear wave equation with nonlinearities of derivative type. By constructing two time‐dependent functionals and employing an iteration method for unbounded multiplier with slicing procedure, the results of blow‐up and upper bound estimates for the lifespan of energy solutions are derived. The model seems to be hyperbolic‐like instead of parabolic‐like. Particularly, the blow‐up result for one‐dimensional case is optimal.

Blow-up and lifespan estimates for a damped wave equation in the Einstein–de Sitter spacetime with nonlinearity of derivative type

In this article, we investigate the blow-up for local solutions to a semilinear wave equation in the generalized Einstein - de Sitter spacetime with nonlinearity of derivative type. More precisely, we consider a semilinear damped wave equation with a time-dependent and not summable speed of propagation and with a time-dependent coefficient for the linear damping term with critical decay rate. We prove in this work that the results obtained in a previous work, where the damping coefficient takes two particular values $0$ or $2$, can be extended for any positive damping coefficient. In the blow-up case, the upper bound of the exponent of the nonlinear term is given, and the lifespan estimate of the global existence time is derived as well.

Sharp lifespan estimates for the weakly coupled system of semilinear damped wave equations in the critical case

The open question, which seems to be also the final part, in terms of studying the Cauchy problem for the weakly coupled system of damped wave equations or reaction-diffusion equations, is so far known as the sharp lifespan estimates in the critical case. In this paper, we mainly investigate lifespan estimates for solutions to the weakly coupled system of semilinear damped wave equations in the critical case. By using a suitable test function method associated with nonlinear differential inequalities, we catch upper bound estimates for the lifespan. Moreover, we establish polynomial-logarithmic type time-weighted Sobolev spaces to obtain lower bound estimates for the lifespan in low spatial dimensions. Then, together with the derived lifespan estimates, new and sharp results on estimates for the lifespan in the critical case are claimed. Finally, we give an application of our results to the semilinear reaction-diffusion system in the critical case.

Continuity of pullback attractors for evolution processes associated with semilinear damped wave equations with time-dependent coefficients

In this paper we consider the semilinear damped wave problem of the form { ( α ( t ) u t ) t − β ( t ) Δ u + γ ( t ) u t + δ ( t ) u = β ( t ) f ( u ) , x ∈ Ω , t > τ , u ( x , t ) = 0 , x ∈ ∂ Ω , t ⩾ τ , u ( x , τ ) = u τ ( x ) , u t ( x , τ ) = v τ ( x ) , x ∈ Ω , where Ω is a bounded smooth domain in R N , N ⩾ 3 , τ ∈ R , f is a real valued function of a real variable with some suitable conditions of growth, regularity and dissipativity, and α , β , γ and δ are continuous real valued functions of a real variable with some suitable conditions of growth, regularity and signs. Using rescaling of time we prove existence, regularity, gradient-like structure, upper and lower semicontinuity of the pullback attractors for the evolution processes associated with this boundary initial value problem in a suitable phase space.

The interplay of critical regularity of nonlinearities in a weakly coupled system of semi-linear damped wave equations

We would like to study a weakly coupled system of semi-linear classical damped wave equations with moduli of continuity in nonlinearities whose powers belong to the critical curve in the p − q plane. The main goal of this paper is to find out sharp conditions of these moduli of continuity which classify between global (in time) existence of small data solutions and finite time blow-up of (even) small data solutions.

Fujita modified exponent for scale invariant damped semilinear wave equations

The aim of this paper is to prove a blow-up result of the solution for a semilinear scale invariant damped wave equation under a suitable decay condition on radial initial data. The admissible range for the power of the nonlinear term depends both on the damping coefficient and on the pointwise decay order of the initial data. In addition, we give an upper bound estimate for the lifespan of the solution. It depends not only on the exponent of the nonlinear term and not only on the damping coefficient but also on the size of the decay rate of the initial data.

Blow-up results for semilinear damped wave equations in Einstein–de Sitter spacetime

We prove by using an iteration argument some blow-up results for a semilinear damped wave equation in generalized Einstein–de Sitter spacetime with a time-dependent coefficient for the damping term and power nonlinearity. Then, we conjecture an expression for the critical exponent due to the main blow-up results, which is consistent with many special cases of the considered model and provides a natural generalization of Strauss exponent. In the critical case, we consider a non-autonomous and parameter dependent Cauchy problem for a linear ODE of second order, whose explicit solutions are determined by means of special functions’ theory.

On the blow – up of solutions to semilinear damped wave equations with power nonlinearity in compact Lie groups

In this note, we prove a blow-up result for the semilinear damped wave equation in a compact Lie group with power nonlinearity |u|p for any p>1, under suitable integral sign assumptions for the initial data, by using an iteration argument. A byproduct of this method is the upper bound estimate for the lifespan of a local in time solution. As a preliminary result, a local (in time) existence result is proved in the energy space via Fourier analysis on compact Lie groups.

On the decay in $ W^{1,\infty} $ for the 1D semilinear damped wave equation on a bounded domain

&lt;p style='text-indent:20px;'&gt;In this paper we study a &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ 2\times2 $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; semilinear hyperbolic system of partial differential equations, which is related to a semilinear wave equation with nonlinear, time-dependent damping in one space dimension. For this problem, we prove a well-posedness result in &lt;inline-formula&gt;&lt;tex-math id="M3"&gt;\begin{document}$ L^\infty $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; in the space-time domain &lt;inline-formula&gt;&lt;tex-math id="M4"&gt;\begin{document}$ (0,1)\times [0,+\infty) $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. Then we address the problem of the time-asymptotic stability of the zero solution and show that, under appropriate conditions, the solution decays to zero at an exponential rate in the space &lt;inline-formula&gt;&lt;tex-math id="M5"&gt;\begin{document}$ L^{\infty} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;. The proofs are based on the analysis of the invariant domain of the unknowns, for which we show a contractive property. These results can yield a decay property in &lt;inline-formula&gt;&lt;tex-math id="M6"&gt;\begin{document}$ W^{1,\infty} $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; for the corresponding solution to the semilinear wave equation.&lt;/p&gt;

Blow-up of solutions to Nakao's problem via an iteration argument

In this paper, we consider blow-up behavior of weak solutions to a weakly coupled system for a semilinear damped wave equation and a semilinear wave equation in Rn. This problem is part of the so-called Nakao's problem proposed by Professor Mitsuhiro Nakao (Kyushu university) for a critical relation between the exponents p and q of the power nonlinearities. By applying an iteration method for unbounded multipliers with a slicing procedure, we prove blow-up of weak solutions for Nakao's problem even for small data. We improve the blow-up result and upper bound estimates for lifespan comparing with the previous research, especially, in higher dimensional cases.