Semilinear Wave Equations Research Articles

Lifespan estimate of solution to the semilinear wave equation with damping term and mass term

<abstract><p>This paper is mainly concerned with the initial boundary value problems of semilinear wave equations with damping term and mass term as well as Neumann boundary conditions on exterior domain in three dimensions. Blow-up and upper bound lifespan estimates of solutions to the problem with damping term and mass term are derived by applying test function technique and iterative method, where nonlinear terms are power nonlinearity $ |u|^p $, derivative nonlinearity $ |u_{t}|^p $, combined nonlinearities $ |u_t|^p+|u|^q $, respectively. Moreover, upper bound lifespan estimate of solution to the problem with scale invariant damping term, non-negative mass term and combined nonlinearities $ |u_t|^p+|u|^q $ is obtained. The proofs are based on the test function method and iterative approach. The main new contribution is that upper bound lifespan estimates of solutions are associated with the Strauss exponent and Glassey exponent. In addition, the variation trend of wave is achieved by taking advantage of numerical simulation.</p></abstract>

AN INVERSE PROBLEM FOR A NONLINEAR WAVE EQUATION WITH DAMPING

We consider an inverse problem of recovering two coefficients in a semi-linear wave equation. This equation contains a damping term and a term with a quadratic nonlinearity. The inverse problem consists in recovering coefficients under these terms as function of the space variable x ∈ R 3 . A forward problem for the equation with a point source is studied. As a result, the inverse problem reduce to two problems, one of them is the well known problem of X-ray tomography, the other one is the problem of the integral geometry with a with a special weight function. The latter problem is studied and a stability estimate for the solution of this problem is stated.

Existence of infinitely many small periodic solutions for a semilinear variable coefficients wave equation with resonant potential

Existence of infinitely many small periodic solutions for a semilinear variable coefficients wave equation with resonant potential

Lifespan estimates of solutions to quasilinear wave equations with damping and negative mass term

<p style='text-indent:20px;'>The main goal of this work is to investigate formation of singularities for solutions to the quasilinear wave equations with damping terms, negative mass terms and divergence form nonlinearities in the critical and sub-critical cases. Upper bound lifespan estimates of solutions are derived by applying the rescaled test function method and iteration technique. The results are the same as corresponding wave equation without damping term and mass term. The main new contribution is that lifespan estimates of solutions are associated with the well-known Strauss exponent and Glassey exponent. To the best of our knowledge, the results in Theorems <inline-formula><tex-math id="M2">\begin{document}$ 1.1-1.4 $\end{document}</tex-math></inline-formula> are new. Moreover, the changing trends of semilinear wave equations are illustrated through numerical simulation.</p>

The blow up of solutions to semilinear wave equations on asymptotically Euclidean manifolds

The blow up of solutions to semilinear wave equations on asymptotically Euclidean manifolds

BLOW-UP OF SOLUTIONS TO THE SEMILINEAR WAVE EQUATIONS WITH FRICTIONAL AND VISCOELASTIC DAMPING TERMS

Our starting point in this paper is to investigate the weakly coupled system of semilinear wave equations with two types of damping terms and combined nonlinearities $|v_{t}|^{p_{1}}+|v|^{q_{1}}$, $|u_{t}|^{p_{2}}+|u|^{q_{2}}$ on exterior domain in $n\,(n\geq1)$ dimensions. Local existence and uniqueness of mild solutions to the problem are established. Moreover, non-existence of global solutions to the problem with power nonlinearities $|u|^{p},\,|v|^{q}$ in the case of coupled system and the problem with power nonlinearity $|u|^{p}$ in the case of single equation are verified. The proofs are based on the test function technique. It is worth noticing that the frictional damping $u_{t}$ is more dominant than the viscoelastic damping $\Delta u_{t}$ when the time trend to infinity. To the best of our knowledge, the blow-up results in Theorems 1.2--1.3 are new.

Constructive exact controls for semi-linear wave equations

Constructive exact controls for semi-linear wave equations

Global existence and nonexistence of solutions for semilinear wave equation with a new condition

In this paper, we consider the initial-boundary problem for semilinear wave equation with a new condition on the source term $ f(u) $ such as \begin{document}$ \begin{equation*} \alpha \int_0^{u } f(s)ds \leq uf(u) + \beta u^2 +\alpha \sigma, \end{equation*} $\end{document} for some positive constants $ \alpha $, $ \beta $, and $ \sigma $, where $ \alpha >2 $ and $ \beta < \frac{\lambda_1(\alpha -2)}{2} $ with $ \lambda_1 $ being a first eigenvalue of Dirichlet-Laplacian. The condition on $ f(u) $ with $ \beta = 0 $ and $ \sigma = 0 $ was studied in [13] and [3]. By introducing a family of potential wells, we are able to establish the invariant sets, vacuum isolation of solutions, and a threshold result of global existence and nonexistence of solutions with the new condition on a source term. Moreover, we discuss the global existence and nonexistence of solutions for problem with initial condition $ E(0)<d $ and critical initial condition $ E(0) = d $.

A competition on blow-up for semilinear wave equations with scale-invariant damping and nonlinear memory term

<p style='text-indent:20px;'>In this paper, we investigate blow-up of solutions to semilinear wave equations with scale-invariant damping and nonlinear memory term in <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R}^n $\end{document}</tex-math></inline-formula>, which can be represented by the Riemann-Liouville fractional integral of order <inline-formula><tex-math id="M2">\begin{document}$ 1-\gamma $\end{document}</tex-math></inline-formula> with <inline-formula><tex-math id="M3">\begin{document}$ \gamma\in(0, 1) $\end{document}</tex-math></inline-formula>. Our main interest is to study mixed influence from damping term and the memory kernel on blow-up conditions for the power of nonlinearity, by using test function method or generalized Kato's type lemma. We find a new competition, particularly for the small value of <inline-formula><tex-math id="M4">\begin{document}$ \gamma $\end{document}</tex-math></inline-formula>, on the blow-up range between the effective case and the non-effective case.</p>

The combined effect in one space dimension beyond the general theory for nonlinear wave equations

In this paper, we show the so-called 'combined effect' of two different kinds of nonlinear terms for semilinear wave equations in one space dimension. This effect means that the lifespan, the maximal existence time, of the classical solution is shorter than the minimum of ones for each term. Such a special phenomenon has been observed in all space dimensions except for one space dimension. We succeed to pick up the combined effect in one space dimension by classifying the lifespan estimates according to whether the total integral of the initial speed is zero, or not, and it is obtained only in the first case. It is also remarkable that, including the combined effect, our results on the lifespan estimates are partially better than those of the general theory for nonlinear wave equations which was considered completed about 30 years ago.

Lifespan estimates for critical semilinear wave equations and scale invariant damped wave equations in exterior domain in high dimensions

Lifespan estimates for critical semilinear wave equations and scale invariant damped wave equations in exterior domain in high dimensions

Lifespan estimates for semilinear wave equations with space dependent damping and potential

In this work, we investigate the influence of general damping and potential terms on the blow-up and lifespan estimates for energy solutions to power-type semilinear wave equations. The space-dependent damping and potential functions are assumed to be critical or short range, spherically symmetric perturbation. The blow up results and the upper bound of lifespan estimates are obtained by the so-called test function method. The key ingredient is to construct special positive solutions to the linear dual problem with the desired asymptotic behavior, which is reduced, in turn, to constructing solutions to certain elliptic “eigenvalue” problems.

Lifespan Estimates for a Class of Semilinear Wave Equations with Time- and Space-Dependent Coefficients on the Power Nonlinearity

Lifespan Estimates for a Class of Semilinear Wave Equations with Time- and Space-Dependent Coefficients on the Power Nonlinearity

Finite-time blowup and existence of global solutions for a logarithmic semilinear hyperbolic equation

In this paper we consider the semilinear wave equation with the product of logarithmic and polynomial nonlinearities and establish the global existence and finite-time blowup of solutions by using the potential well method.

Stabilization of semilinear wave equations with time-dependent variable coefficients and memory

Stabilization of semilinear wave equations with time-dependent variable coefficients and memory

On a Semilinear Wave Equation with Kirchhoff-Type Nonlocal Damping Terms and Logarithmic Nonlinearity

On a Semilinear Wave Equation with Kirchhoff-Type Nonlocal Damping Terms and Logarithmic Nonlinearity

Global existence for a system of multiple-speed wave equations violating the null condition

We discuss the Cauchy problem for a system of semilinear wave equations in three space dimensions with multiple wave speeds. Though our system does not satisfy the standard null condition, we show that it admits a unique global solution for any small and smooth data. This generalizes a preceding result due to Pusateri and Shatah. The proof is carried out by the energy method involving a collection of generalized derivatives. The multiple wave speeds disable the use of the Lorentz boost operators, and our proof therefore relies upon the version of Klainerman and Sideris. Due to the presence of nonlinear terms violating the standard null condition, some of components of the solution may have a weaker decay as $t\to\infty$, which makes it difficult even to establish a mildly growing (in time) bound for the high energy estimate. We overcome this difficulty by relying upon the ghost weight energy estimate of Alinhac and the Keel-Smith-Sogge type $L^2$ weighted space-time estimate for derivatives.

Newton's method for stochastic semilinear wave equations driven by multiplicative time‐space noise

AbstractSemilinear wave equations with additive or one‐dimensional noise are treatable by various iterative and numerical methods. We study more difficult models of semilinear wave equations with infinite dimensional multiplicative spatially correlated noise. Our proof of probabilistic second‐order convergence of some iterative methods is based on Da Prato and Zabczyk's maximal inequalities.

Classification of the Blow-Up Behavior for a Semilinear Wave Equation with Nonconstant Degenerate Coefficients

We consider a nonlinear wave equation with nonconstant coefficients. In particular, the coefficient in front of the second-order space derivative is degenerate. We give the blow-up behavior and the regularity of the blow-up set. Partial results are given at the origin, where the degeneracy occurs. Some nontrivial obstacles, due to the nonconstant speed of propagation, have to be surmounted.

On a strongly damped semilinear wave equation with time-varying source and singular dissipation

Abstract This paper deals with the global well-posedness and blow-up phenomena for a strongly damped semilinear wave equation with time-varying source and singular dissipative terms under the null Dirichlet boundary condition. On the basis of cut-off technique, multiplier method, contraction mapping principle, and the modified potential well method, we establish the local well-posedness and obtain the threshold between the existence and nonexistence of the global solution (including the critical case). Meanwhile, with the aid of modified differential inequality technique, the blow-up result of the solutions with arbitrarily positive initial energy and the lifespan of the blow-up solutions are derived.