Stochastic Wave Equations with Polynomial Nonlinearity

Chapter 7 On the Global Weak Solutions to a Variational Wave Equation

The paper is concerned with the problem of explosive solutions for a class of nonlinear stochastic wave equations in a domain $\mathcal{D}\subset\mathbb{R}^{d}$ for d≤3. Under appropriate conditions on the initial data, the nonlinear term and the noise intensity is proved in Theorem 3.1 that the L2-norm of the solution will blow up at a finite time in the mean-square sense. An example is given to show an application of the theorem.

This paper is concerned with the lifespan of and the blowup mechanism for smooth solutions to the 2-D nonlinear wave equation ∂ t 2 u − ∑ i = 1 2 ∂ i ( c i 2 ( u ) ∂ i u ) \partial _t^2u-\sum _{i=1}^2\partial _i(c_i^2(u)\partial _iu) = 0 =0 , where c i ( u ) ∈ C ∞ ( R n ) c_i(u)\in C^{\infty }(\mathbb {R}^n) , c i ( 0 ) ≠ 0 c_i(0)\neq 0 , and ( c 1 ′ ( 0 ) ) 2 + ( c 2 ′ ( 0 ) ) 2 ≠ 0 (c_1’(0))^2+(c_2’(0))^2\neq 0 . This equation has an interesting physical background as it arises from the pressure-gradient model in compressible fluid dynamics and also in nonlinear variational wave equations. Under the initial condition ( u ( 0 , x ) , ∂ t u ( 0 , x ) ) = ( ε u 0 ( x ) , ε u 1 ( x ) ) (u(0,x), \partial _tu(0,x))=(\varepsilon u_0(x), \varepsilon u_1(x)) with u 0 ( x ) , u 1 ( x ) ∈ C 0 ∞ ( R 2 ) u_0(x), u_1(x)\in C_0^{\infty }(\mathbb {R}^2) , and ε > 0 \varepsilon >0 is small, we will show that the classical solution u ( t , x ) u(t,x) stops to be smooth at some finite time T ε T_{\varepsilon } . Moreover, blowup occurs due to the formation of a singularity of the first-order derivatives ∇ t , x u ( t , x ) \nabla _{t,x}u(t,x) , while u ( t , x ) u(t,x) itself is continuous up to the blowup time T ε T_{\varepsilon } .

In this article we study the mean square consistency on numerical solutions of stochastic wave equations with cubic nonlinearities on two dimensional rectangles. In [8], we proved that the strong Fourier solution of these semi-linear wave equations exists and is unique on an appropriate Hilbert space. A linear-implicit Euler method is used to discretize the related Fourier coefficients. We prove that the linear-implicit Euler method applied to a solution of nonlinear stochastic wave equations in two dimensions is mean square consistency under the geometric condition.

Nonlinear stochastic wave equation driven by rough noise

A characterization of global and nonglobal solutions of nonlinear wave and Kirchhoff equations

The exact controllability for a semilinear stochastic wave equation with a boundary control is established. The target and initial spaces are L2(G) × H−1(G) with G being a bounded open subset of R3 and the nonlinear terms having at most a linear growth.

A corrected derivation of nonlinear wave propagation equations with fractional loss operators is presented. The fundamental approach is based on fractional formulations of the stress-strain and heat flux definitions but uses the energy equation and thermodynamic identities to link density and pressure instead of an erroneous fractional form of the entropy equation as done in Prieur and Holm ["Nonlinear acoustic wave equations with fractional loss operators," J. Acoust. Soc. Am. 130(3), 1125-1132 (2011)]. The loss operator of the obtained nonlinear wave equations differs from the previous derivations as well as the dispersion equation, but when approximating for low frequencies the expressions for the frequency dependent attenuation and velocity dispersion remain unchanged.

Explosion of solutions to nonlinear stochastic wave equations with multiplicative noise

PurposeThis paper aims to solve linear and non-linear shallow water wave equations using homotopy perturbation method (HPM). HPM is a straightforward method to handle linear and non-linear differential equations. As such here, one-dimensional shallow water wave equations have been considered to solve those by HPM. Interesting results are reported when the solutions of linear and non-linear equations are compared.Design/methodology/approachHPM was used in this study.FindingsSolution of one-dimensional linear and non-linear shallow water wave equations and comparison of linear and non-linear coupled shallow water waves from the results obtained using present method.Originality/valueCoupled non-linear shallow water wave equations are solved.

On some nonlinear hyperbolic systems with damping boundary conditions

This paper concerns with the global classical solution to the Cauchy problem of the nonlinear double dispersive wave equation with strong damping uttuutt + � 2 u cut = d N X i=1 ∂ ∂xi σ i(uxi), where c and d are positive constants. By the contraction mapping principle and priori estimates, we prove that the Cauchy problem admits a unique global classical solution, and by the concavity method, we give the sufficient condi- tions on the blowup of the global solution of the Cauchy problem. Finally, as an application, an example is also given.

Consider a nonlinear stochastic wave equation driven by space-time white noise in dimension one. We discuss the intermittency of the solution, and then use those intermittency results in order to demonstrate that in many cases the solution is chaotic. For the most part, the novel portion of our work is about the two cases where (1) the initial conditions have compact support, where the global maximum of the solution remains bounded, and (2) the initial conditions are positive constants, where the global maximum is almost surely infinite. Bounds are also provided on the behavior of the global maximum of the solution in Case (2).KeywordsIntermittencyThe stochastic wave equationChaos

The boundness of the operator-valued functions for multidimensional nonlinear wave equations with applications

We prove existence of global solutions to a semilinear massless Dirac system with small initial data. We study solutions in generalised Sobolev spaces suggested by S. Klainerman. Our approach is based on using conservation law of charge together with Sobolev type weighted estimates for the spinor field. Our result seems to be sharp in a view of blowing-up results obtained by F. John (see [7]). We also study decay properties of the spinor field. With similar methods we prove global existence for a nonlinear wave equation in three space dimension. The same equation was studied by T. Sideris [14] and H. Takamura [15]. They proved global existence for spherically symmetrical initial data. In this work we remove this condition on the initial data.