Global existence for time-dependent damped wave equations with nonlinear memory

Global space-time Trefftz DG schemes for the time-dependent linear wave equation

It has been previously shown [Horace G. Ferris, “Computation of Farfield Radiation Patterns by Use of a General Integral Solution to the Time Dependent Scalar Wave Equation,” J. Acoust. Soc. Amer. 41, 394–400 (1967)] that the formulation of the farfield pressure in terms of an integral formula proceeded from the time-dependent scalar wave equation and thus had no single-frequency assumption. Thus, the farfield prediction theory should be valid for broad-band, as well as single-frequency signals. Measurements made by Hughes Aircraft Company and others have demonstrated the validity of the prediction theory, as applied to single-frequency steady-state signals. It has remained unverified, however, in the realm of time-dependent broadband signals. Measurements have been made in the Hughes Test Tank driving a narrow-bandwidth transducer with white noise. The pulse shape, as measured in the farfield has been accurately reproduced from measurements made over a small surface in the nearfield of the source. No attempts were made to predict the magnitude since such a small portion of the nearfield was covered. This suggests that the theory can be applied to the prediction of broad-band signals from measurements made in the nearfield.

This paper is concerned with the Cauchy problem for one-dimensional unipolar Euler-Poisson equations with time-dependent damping, where the time-asymptotically degenerate damping in the form of - (1+t) u for > 0 with > 0 plays a crucial role for the structure of solutions. The main issue of the paper is to investigate the critical case with = 1. We first prove that, for all cases with > 0 and > 0 (including the critical case of = 1), once the initial data is steep at a point, then the solutions are locally bounded but their derivatives will blow up in finite time, by means of the method of Riemann invariants and the technical convex analysis. Secondly, for the critical case of = 1 with > 7/3, we prove that there exists a unique global solution, once the initial perturbation around the constant steady-state is sufficiently small. In particular, we derive the algebraic convergence rates of the solution to the constant steady-state, which are piecewise, related to the parameter for 7/3 < 3, 3 < 4 and > 4. The adopted method of proof in this critical case is the technical time-weighted energy method and the time-weight depends on the parameter . Finally, we carry out some numerical simulations in two cases for blow-up and global existence, respectively, which numerically confirm our theoretical results.

Hadamard integrator for time-dependent wave equations: Lagrangian formulation via ray tracing

Blow-up analysis for a quasilinear parabolic system with multi-coupled nonlinearities

The quantum mechanical, time-dependent wave equation in generalized coordinates is obtained by a slight modification of Schrödinger's original derivation. If the classical Hamiltonian for a system is known, the variational technique may be applied to the Hamilton-Jacobi equation to yield an Euler-Lagrange equation. This latter is then the wave equation for the system. The method, which is generally easy to apply, is illustrated by applications to a single particle in curvilinear coordinates and to a symmetric top molecule.

This chapter is devoted to the classical theory of the reduced wave equation in domains containing a neighborhood of infinity. The main difference with respect to the standard vibration problem of Chapter I is that the imbedding H1 ⊂ L2 is no longer compact. The spectrum of the Laplace operator (with standard boundary conditions) is purely continuous. The corresponding spectral family is considered in Section 5. The solution of the nonhomogeneous boundary value problems requires a radiation condition (Sections 3 and 4). These solutions describe the limit state as t → + ∞ of the corresponding time-dependent wave equation. The radiation of energy towards infinity implies some decay of solutions as t → +∞. There exist (complex) scattering frequencies ω s which are some sort of eigenvalues for the vibration (Sections 3, 4, and 8). The relation with the corresponding time-dependent vibration (which exhibits wave fronts) is explained in Section 2. Most of the material of this chapter is presented in the case of space dimension 3, but certain results hold true in the more difficult case of dimension 2.

In this article we consider wave equations with constant and linear time dependent diffusion-coefficients which are solved numerically by iterative operator splitting with interval overlapping algorithms. The benefits of overlappling for time dependent equations are discussed. A splitting analysis and the assemblation are presented. Numerical examples for 2D wave equations are discussed at the end of this paper. Keyword numerical analysis, operator-splitting method, initial value problems, iterative solver method, stability analysis, overlapping schemes, convectiondiffusion-reaction equation. AMS subject classifications. 35J60, 35J65, 65M99, 65N12, 65Z05, 74S10, 76R50.

Mixed-dimensional coupling for time-dependent wave problems using the Nitsche method

A new numerical approach for the time-dependent wave and heat equations as well as for the time-independent Poisson equation on irregular domains has been developed. Trivial Cartesian meshes and simple 9-point stencil equations with unknown coefficients are used for 2-D irregular domains. The calculation of the coefficients of the stencil equations is based on the minimization of the local truncation error of the stencil equations and yields the optimal order of accuracy. The treatment of the Dirichlet and Neumann boundary conditions in the new approach is related to the development of high-order boundary conditions with the stencils that include the same or a smaller number of grid points compared to that for the regular 9-point internal stencils. At similar 9-point stencils, the accuracy of the new approach is two orders higher than that for the linear finite elements. The numerical results for irregular domains in Part 2 of the paper also show that at the same number of degrees of freedom, the new approach is even much more accurate than the quadratic and cubic finite elements with much wider stencils. Similar to our recent results on regular domains, the order of the accuracy of the new approach for the Poisson equation on irregular domains with square Cartesian meshes is higher than that with rectangular Cartesian meshes. The new approach can be directly applied to other partial differential equations.

In this paper, we consider the initial boundary value problem for a class of fourth-order wave equations with strong damping term, nonlinear weak damping term, strain term and nonlinear source term in polynomial form. First, the local solution is obtained by using fix point theory. Then, by constructing the potential well structure frame, we get the global existence, asymptotic behavior and blowup of solutions for the subcritical initial energy and critical initial energy respectively. Ultimately, we prove the blowup in finite time of solutions for the arbitrarily positive initial energy case.

In this paper we use a modified test function method to derive nonexistence results for the semilinear wave equation with time-dependent speed and damping. The obtained critical exponent is the same exponent of some recent results on global existence of small data solutions.

Small data blow-up for the wave equation with a time-dependent scale invariant damping and a cubic convolution for slowly decaying initial data

In this article, we deal with the global existence and nonexistence of solutions to the non-Newtonian polytropic filtration equations coupled with nonlinear boundary conditions. By constructing various kinds of sub- and super-solutions and using the basic properties of M-matrix, we give the necessary and sufficient conditions for global existence of nonnegative solutions. The critical curve of Fujita type is conjectured with the aid of some new results, which extend the recent results of Zheng, Song, and Jiang [Critical Fujita exponents for degenerate parabolic equations coupled via nonlinear boundary flux, J. Math. Anal. Appl. 298 (2004), pp. 308–324], Zhou and Mu [Critical curve for a non-Newtonian polytropic filtration system coupled via nonlinear boundary flux, Nonlinear Anal. 68 (2008), pp. 1–11], and Zhou and Mu [Algebraic criteria for global existence or blow-up for a boundary coupled system of nonlinear diffusion equations, Appl. Anal. 86 (2007), pp. 1185–1197] to more general equations.

We consider the Gierer-Meinhardt system (1.1), shown below, on a bounded smooth domain $\Omega\subset\mathbb{R}^n$ ($n\ge1$) with a homogeneous Neumann boundary condition. For suitable exponents $a$, $b$, $c$ and $d$, we establish certain sufficient conditions for global existence. Theorem 1.1 here, combined with Theorem 1.2 of [6], implies a classical phenomenonon the effect of the initial data on global existence and finite time blow-up. This work is a continuation of our earlier result [6] for the Gierer-Meinhardt system.&nbsp; &nbspThe Gierer-Meinhardt system was introduced in [1] to model activator-inhibitor systems in pattern formation in ecological systems.