Semilinear Wave Equations Research Articles

On initial-boundary value problems for one-dimension semilinear wave equations with null conditions

We consider the initial-boundary value problem on R+×R+ for one-dimension systems of semilinear wave equations with null conditions. We show that for homogeneous Dirichlet or Neumann boundary values and sufficiently small initial data, classical solutions always globally exist. Furthermore, we also prove that the global solution is asymptotically free in the energy sense.

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.

Decay of semilinear damped wave equations: cases without geometric control condition

We consider the semilinear damped wave equation $\partial^2_{tt} u(x,t) + \gamma(x)\partial_t u(x,t) = \Delta u(x,t) - \alpha u(x,t) - f (x,u(x,t))$. In this article, we obtain the first results concerning the stabilization of this semilinear equation in cases where $\gamma$ does not satisfy the geometric control condition. When some of the geodesic rays are trapped, the stabilization of the linear semigroup is semi-uniform in the sense that $\|e^{At}A^{-1}\| \leq h(t)$ for some function $h$ with $h(t)\rightarrow 0$ when $t\rightarrow +\infty$. We provide general tools to deal with the semilinear stabilization problem in the case where $h(t)$ has a sufficiently fast decay.

Weak convergence of fully discrete finite element approximations of semilinear hyperbolic SPDE with additive noise

The numerical approximation of the mild solution to a semilinear stochastic wave equation driven by additive noise is considered. A standard finite element method is employed for the spatial approximation and a a rational approximation of the exponential function for the temporal approximation. First, strong convergence of this approximation in both positive and negative order norms is proven. With the help of Malliavin calculus techniques this result is then used to deduce weak convergence rates for the class of twice continuously differentiable test functions with polynomially bounded derivatives. Under appropriate assumptions on the parameters of the equation, the weak rate is found to be essentially twice the strong rate. This extends earlier work by one of the authors to the semilinear setting. Numerical simulations illustrate the theoretical results.

Nonexistence of global solutions for a weakly coupled system of semilinear damped wave equations in the scattering case with mixed nonlinear terms

In this paper we consider the blow-up of solutions to a weakly coupled system of semilinear damped wave equations in the scattering case with nonlinearities of mixed type. The proof of the blow-up results is based on an iteration argument. We find as critical curve for the pair of exponents (p, q) in the nonlinear terms the same one found for the weakly coupled system of semilinear wave equations with the same kind of nonlinearities. In the critical and not-damped case we combine an iteration argument with the so-called slicing method to show the blow-up dynamic of a weighted version of the functionals used in the subcritical case.

Blow-up for a coupled system of semilinear wave equations with scattering dampings and combined nonlinearities

The blow-up of solutions to a coupled system of semilinear wave equations in the sub-critical and critical cases is investigated. The system contains scattering dampings and combined nonlinearities. The upper bound lifespan estimates of solutions which are related to the indexes of nonlinear terms are established by applying the functional method and iteration technique.

Regular generalized solutions to semilinear wave equations

The paper is devoted to proving an existence and uniqueness result for generalized solutions to semilinear wave equations with a small nonlinearity in space dimensions 1, 2, 3. The setting is the one of Colombeau algebras of generalized functions. It is shown that for a nonlinearity of arbitrary growth and sign, but multiplied with a small parameter, the initial value problem for the semilinear wave equation has a unique solution in the Colombeau algebra of generalized functions of bounded type. The proof relies on a fixed point theorem in the ultra-metric topology on the algebras involved. In classical terms, the result says that the semilinear wave equations under consideration have global classical solutions up to a rapidly vanishing error.

A reaction-diffusion approximation of a semilinear wave equation

A semilinear wave equation possesses propagation with a finite speed, while a parabolic equation has propagation with infinite speed. This paper proposes a reaction-diffusion system whose solutions approximate those of a semilinear wave equation under some assumptions of a reaction term. The proof is based on the energy method.

Interplay effects on blow-up of weakly coupled systems for semilinear wave equations with general nonlinear memory terms

In this paper, we study weakly coupled systems for semilinear wave equations with distinct nonlinear memory terms in general forms, and the corresponding single semilinear equation with general nonlinear memory terms. Thanks to fixed point theory, we prove local (in time) existence of solutions with the L1 assumption on the memory kernels. Then, blow-up results for energy solutions are derived by applying iteration methods associated with slicing procedure. We investigate interactions on the blow-up conditions under different decreasing assumptions on the memory term. Particularly, a new threshold for the kernels on interplay effect is found. Additionally, we give some applications of our results on semilinear wave equations and acoustic wave equations.

Regularity of Invariant Sets in Variable Internal Damped Wave Equations

In this paper we prove that every compact invariant subset $$\mathscr{A}$$ associated with the semigroup Sn,k(t)t≥0 generated by wave equations with variable damping, either in the interior or on the boundary of the domain where Ω ⊂ ℝ3 is a smooth bounded domain, in H 0 1 (Ω) × L2(Ω) is in fact bounded in D(B0) × H 0 1 (Ω). As an application of our results, we obtain the upper-semicontinuity for global attractor of the weakly damped semilinear wave equation in the norm of H1(Ω) × L2(Ω) when the interior variable damping converges to the boundary damping in the sense of distributions.

Strong Exponential Attractors for Weakly Damped Semilinear Wave Equations

In this paper, we investigate the longtime dynamics for the damped wave equation in a bounded smooth domain of ℝ3. The exponential attractor is investigated in a strong energy space for the case of subquintic nonlinearity, which is based on the recent extension of the Strichartz estimate for the case of a bounded domain. The results obtained complete some previous works.

Hadamard well-posedness for a structure acoustic model with a supercritical source and damping terms

<p style='text-indent:20px;'>This article is concerned with Hadamard's well posedness of a structural acoustic model consisting of a semilinear wave equation defined on a smooth bounded domain <inline-formula><tex-math id="M1">\begin{document}$ \Omega\subset\mathbb{R}^3 $\end{document}</tex-math></inline-formula> which is strongly coupled with a Berger plate equation acting only on a flat part of the boundary of <inline-formula><tex-math id="M2">\begin{document}$ \Omega $\end{document}</tex-math></inline-formula>. The system is influenced by several competing forces. In particular, the source term acting on the wave equation is allowed to have a <i>supercritical</i> exponent, in the sense that its associated Nemytskii operators is not locally Lipschitz from <inline-formula><tex-math id="M3">\begin{document}$ H^1_{\Gamma_0}(\Omega) $\end{document}</tex-math></inline-formula> into <inline-formula><tex-math id="M4">\begin{document}$ L^2(\Omega) $\end{document}</tex-math></inline-formula>. By employing nonlinear semigroups and the theory of monotone operators, we obtain several results on the existence of local and global weak solutions. Moreover, we prove that such solutions depend continuously on the initial data, and uniqueness is obtained in two different scenarios.</p>

Heat-like and wave-like lifespan estimates for solutions of semilinear damped wave equations via a Kato's type lemma

In this work we consider several semilinear damped wave equations with “subcritical” nonlinearities, focusing on studying lifespan estimates for energy solutions. Our main concern is on equations with scale-invariant damping and mass. By imposing different assumptions on the initial data, we prove lifespan estimates from above, distinguishing between “wave-like” and “heat-like” behaviours. Furthermore, we conjecture logarithmic improvements for the estimates on the “transition surfaces” separating the two behaviours. As a direct consequence, we reorganize the blow-up results and lifespan estimates for the massless case, and we obtain in particular improved lifespan estimates for the one dimensional case, compared to the known results.We also study semilinear wave equations with scattering damping and negative mass term, finding that if the decay rate of the mass term equals to 2, the lifespan estimate coincides with the one in a special case of scale-invariant damped equation.The main tool employed in the proof is a Kato's type lemma, established by iteration argument.

On Agemi-type structural conditions for a system of semilinear wave equations

We consider a two-component system of cubic semilinear wave equations in two space dimensions satisfying the Agemi-type structural condition (Ag) but violating (Ag0) and (Ag[Formula: see text]). For this system, we show that small amplitude solutions are asymptotically free as [Formula: see text].

Blow up of solutions of semilinear wave equations in Friedmann–Lemaître–Robertson–Walker spacetime

Consider a nonlinear wave equation for a massless scalar field with self-interaction in the spatially flat Friedmann–Lemaître–Robertson–Walker spacetimes. For the case of decelerating expansion, we show upper bounds of the lifespan of blow-up solutions by distinguishing subcritical and critical cases. Comparing to the case of the Minkowski spacetime, we discuss how the scale factor affects the lifespan of blow-up solutions of the equation.

Inward/outward energy theory of non-radial solutions to 3D semi-linear wave equation

The topic of this paper is a semi-linear, energy sub-critical, defocusing wave equation ∂t2u−Δu=−|u|p−1u in the 3-dimensional space with 3≤p<5. We generalize inward/outward energy theory and weighted Morawetz estimates for radial solutions to the non-radial case. As an application we show that if 3<p<5 and κ>5−p2, then the solution scatters as long as the initial data (u0,u1) satisfy∫R3(|x|κ+1)(12|∇u0|2+12|u1|2+1p+1|u0|p+1)dx<+∞. If p=3, we can also prove the scattering result if initial data (u0,u1) are contained in the critical Sobolev space and satisfy the inequality∫R3|x|(12|∇u0|2+12|u1|2+14|u0|p+1)dx<+∞. These assumptions on the decay rate of initial data as |x|→∞ are weaker than previously known scattering results in the non-radial case.

A condition for blow-up solutions to discrete semilinear wave equations on networks

In this paper, we consider discrete semilinear wave equations where S is a network with boundary . The main contribution of this work is to introduce a condition for some and with , where is the first eigenvalue of discrete Laplacian , and we use the concavity method to obtain the blow-up solutions to the above equations.

Decay rates for semilinear wave equations with vanishing damping and Neumann boundary conditions

The paper is concerned with the semilinear wave equations with time‐dependent damping γ(t)=α/(1+t) (α&gt;0), under the effect of nonlinear source f behaving like a polynomial, and subject to Neumann boundary conditions. Constructing appropriate auxiliary functions, we obtain an explicit uniform decay rate estimate for the solutions of the equation in terms of the exponent of f, when α is large enough. On the other hand, via a new hyperbolic version of Dirichlet quotients, we show that the upper estimate is optimal in some case, which implies the existence of slow solutions.

Propagation of singularities for generalized solutions to nonlinear wave equations

The paper is devoted to regularity theory of generalized solutions to semilinear wave equations with a small nonlinearity. The setting is the one of Colombeau algebras of generalized functions. It is shown that in one space dimension, an initial singularity at the origin propagates along the characteristic lines emanating from the origin, as in the linear case. The proof is based on a fixed point theorem in a suitable ultrametric topology on the subset of Colombeau solutions possessing the required regularity. The paper takes up the initiating research of the 1970s on anomalous singularities in classical solutions to semilinear hyperbolic equations, and shows that the same behavior is attained in the case of non-classical, generalized solutions.

Solvability of a Mixed Problem with Nonlinear Boundary Condition for a One-Dimensional Semilinear Wave Equation

For a one-dimensional semilinear wave equation, a mixed problem with nonlinear boundary condition is considered. The uniqueness and the local and global solvability of the problem under consideration are studied depending on the type of nonlinearities in the equation and in the boundary conditions. The cases of nonexistence of a solution not only globally but even locally are considered, as well as the case where this problem has a blow-up solution.