Wave Equations In Higher Dimensions Research Articles

Long time behaviour of finite-energy radial solutions to energy subcritical wave equation in higher dimensions

We consider the defocusing, energy subcritical wave equation ∂t2u−Δu=−|u|p−1u in 4 to 6 dimensional spaces with radial initial data. We define w=r(d−1)/2u, reduce the equation above to one-dimensional equation of w and apply method of characteristic lines. This gives scattering of solutions outside any given light cone as long as the energy is finite. The scattering in the whole space can also be proved if we assume the energy decays at a certain rate as x→+∞. This generalizes the 3-dimensional results in Shen [32] to higher dimensions.

Improved local smoothing estimate for the wave equation in higher dimensions

In this paper, we establish the sharp k-broad estimate for a class of phase functions satisfying the homogeneous convex conditions. As an application, we obtain improved local smoothing estimates for the half-wave operator in dimensions n≥3. As a byproduct, we also generalize the restriction estimates of Ou–Wang [21] to a broader class of phase functions.

The blow-up rate for a non-scaling invariant semilinear wave equations in higher dimensions

We consider the semilinear wave equation ∂t2u−Δu=f(u),(x,t)∈RN×[0,T),(1) with f(u)=|u|p−1uloga(2+u2), where p>1 and a∈R, with subconformal power nonlinearity. We will show that the blow-up rate of any singular solution of (1) is given by the ODE solution associated with (1), The result in one space dimension, has been proved in Hamza and Zaag (2020). Our goal here is to extend this result to higher dimensions.

Stable blowup for the cubic wave equation in higher dimensions

We consider the wave equation with a focusing cubic nonlinearity in higher odd space dimensions without symmetry restrictions on the data. We prove that there exists an open set of initial data such that the corresponding solution exists in a backward light-cone and approaches the ODE blowup profile.

Wave equation in higher dimensions - periodic solutions

Wave equation in higher dimensions - periodic solutions

Integral representation for solutions of the wave equation by shearlets

We build an integral representation of solutions of the wave equation as a superposition of other solutions. Investigations are based on continuous shearlet transforms in higher dimensions. We construct a new shearlet which helps us form a class of solutions for the wave equation in higher dimensions.

Dynamics near explicit stationary solutions in similarity variables for solutions of a semilinear wave equation in higher dimensions

This is the first of two papers devoted to the study of the properties of the blow-up surface for the N N dimensional semilinear wave equation with subconformal power nonlinearity. In a series of papers, we have clarified the situation in one space dimension. Our goal here is to extend some of the properties to higher dimension. In dimension one, an essential tool was to study the dynamics of the solution in similarity variables, near the set of non-zero equilibria, which are obtained by a Lorentz transform of the space-independent solution. As a matter of fact, the main part of this paper is to study similar objects in higher dimensions. More precisely, near that set of equilibria, we show that solutions are either non-global or go to zero or converge to some explicit equilibrium. We also show that the first case cannot occur in the characteristic case and that only the third possibility occurs in the non-characteristic case, thanks to the non-degeneracy of the blow-up limit, another new result in our paper. As a by-product of our techniques, we obtain the stability of the zero solution.

An analytical expression of solutions to nonlinear wave equations in higher dimensions with Robin boundary conditions

It is known that an analytical expression of solutions to nonlinear wave equations in higher dimensions may be helpful to recognizing new nonlinear physical phenomena and developing novel numerical integrators. Therefore, in this paper, we present an analytical expression for the initial–boundary value problem of higher-dimensional nonlinear wave equations Utt(X,t)−a2ΔU(X,t)=f(U(X,t),Ut(X,t)), U(X,t0)=U0(X), Ut(X,t0)=U1(X), posed on a bounded domain Ω⊂Rd for d≥1 and equipped with the Robin boundary conditions ∇U⋅n+λU=β(X,t), X∈∂Ω, where n is the unit outward normal vector at the boundary ∂Ω, and λ is a constant.

An elementary proof of Strauss conjecture

In this paper, we consider the Cauchy problem for semilinear wave equation in high dimensions. First we use a Morawetz energy estimate which is obtained by integrating on the light cone to get a weighted L2−L2 estimate of the solution, and then give an elementary proof of the weighted Strichartz estimate in Georgiev et al. [3], hence the Strauss conjecture. We also obtain a variant of the weighted Strichartz estimates and give the sharp estimate of the lifespan for the semilinear wave equation with subcritical nonlinearity.

Stability and Unconditional Uniqueness of Solutions for Energy Critical Wave Equations in High Dimensions

In this paper we establish a complete local theory for the energy-critical nonlinear wave equation (NLW) in high dimensions ℝ × ℝ d with d ≥ 6. We prove the stability of solutions under the weak condition that the perturbation of the linear flow is small in certain space-time norms. As a by-product of our stability analysis, we also prove local well-posedness of solutions for which we only assume the smallness of the linear evolution. These results provide essential technical tools that can be applied towards obtaining the extension to high dimensions of the analysis of Kenig and Merle [17] of the dynamics of the focusing (NLW) below the energy threshold. By employing refined paraproduct estimates we also prove unconditional uniqueness of solutions for d ≥ 6 in the natural energy class. This extends an earlier result by Planchon [26].

Blow-up results for semilinear wave equations in the superconformal case

We consider the semilinear wave equation in higher dimensions with power nonlinearity in the superconformal range, and its perturbations with lower order terms, including the Klein-Gordon equation. We improve the upper bounds on blow-up solutions previously obtained by Killip, Stovall and Visan [22]. Our proof uses the similarity variables' setting. We consider the equation in that setting as a perturbation of the conformal case, and we handle the extra terms thanks to the ideas we already developed in [16] for perturbations of the pure power conformal case with lower order terms.

The stochastic wave equation in high dimensions: Malliavin differentiability and absolute continuity

We consider the class of non-linear stochastic partial differential equations studied in [Conus-Dalang, 2008]. Equivalent formulations using integration with respect to a cylindrical Brownian motion and also the Skorohod integral are established. It is proved that the random field solution to these equations at any fixed point $(t,x)\in[0,T]\times \mathbb{R}^d$ is differentiable in the Malliavin sense. For this, an extension of the integration theory in [Conus-Dalang, 2008] to Hilbert space valued integrands is developed, and commutation formulae of the Malliavin derivative and stochastic and pathwise integrals are proved. In the particular case of equations with additive noise, we establish the existence of density for the law of the solution at $(t,x)\in]0,T]\times\mathbb{R}^d$. The results apply to the stochastic wave equation in spatial dimension $d\ge 4$.

The sharp upper bound of the lifespan of solutions to critical semilinear wave equations in high dimensions

The final open part of Straussʼ conjecture on semilinear wave equations was the blow-up theorem for the critical case in high dimensions. This problem was solved by Yordanov and Zhang (2006) [18], or Zhou (2007) [21] independently. But the estimate for the lifespan, the maximal existence time, of solutions was not clarified in both papers. In this paper, we refine their theorems and introduce a new iteration argument to get the sharp upper bound of the lifespan. As a result, with the sharp lower bound by Li and Zhou (1995) [10], the lifespan T ( ε ) of solutions of u t t − Δ u = u 2 in R 4 × [ 0 , ∞ ) with the initial data u ( x , 0 ) = ε f ( x ) , u t ( x , 0 ) = ε g ( x ) of a small parameter ε > 0 , compactly supported smooth functions f and g, has an estimate exp ( c ε − 2 ) ⩽ T ( ε ) ⩽ exp ( C ε − 2 ) , where c and C are positive constants depending only on f and g. This upper bound has been known to be the last open optimality of the general theory for fully nonlinear wave equations.

Radial Fourier multipliers in high dimensions

Given a fixed p≠2, we prove a simple and effective characterization of all radial multipliers of $ \mathcal{F}{L^p}\left( {{\mathbb{R}^d}} \right) $, provided that the dimension d is sufficiently large. The method also yields new Lq space-time regularity results for solutions of the wave equation in high dimensions.

Dynamics for the energy critical nonlinear wave equation in high dimensions

T. Duyckaerts and F. Merle (2008) studied the variational structure near the ground state solution W W of the energy critical wave equation and classified the solutions with the threshold energy E ( W , 0 ) E(W,0) in dimensions d = 3 , 4 , 5 d=3,4,5 . In this paper, we extend the results to all dimensions d ≥ 6 d\ge 6 . The main issue in high dimensions is the non-Lipschitz continuity of the nonlinearity which we get around by making full use of the decay property of W W .

Boundary Controllability for the Quasilinear Wave Equation

We study the boundary exact controllability for the quasilinear wave equation in high dimensions. Our main tool is the geometric analysis. We derive the existence of long time solutions near an equilibrium, prove the locally exact controllability around the equilibrium under some checkable geometrical conditions. We then establish the globally exact controllability in such a way that the state of the quasilinear wave equation moves from an equilibrium in one location to an equilibrium in another location under some geometrical conditions. The Dirichlet action and the Neumann action are studied, respectively. Our results show that exact controllability is geometrical characters of a Riemannian metric, given by the coefficients and equilibria of the quasilinear wave equation. A criterion of exact controllability is given, which based on the sectional curvature of the Riemann metric. Some examples are presented to verify the global exact controllability.

Optimal Discontinuous Galerkin Methods for the Acoustic Wave Equation in Higher Dimensions

In this paper, we develop and analyze a new class of discontinuous Galerkin (DG) methods for the acoustic wave equation in mixed form. Traditional mixed finite element (FE) methods produce energy conserving schemes, but these schemes are implicit, making the time-stepping inefficient. Standard DG methods give explicit schemes, but these approaches are typically dissipative or suboptimally convergent, depending on the choice of numerical fluxes. Our new method can be seen as a compromise between these two kinds of techniques, in the way that it is both explicit and energy conserving, locally and globally. Moreover, it can be seen as a generalized version of the Raviart–Thomas FE method and the finite volume method. Stability and convergence of the new method are rigorously analyzed, and we have shown that the method is optimally convergent. Furthermore, in order to apply the new method for unbounded domains, we apply our new method with the first order absorbing boundary condition. The stability of the resulting numerical scheme is analyzed.

Low-frequency dispersive estimates for the wave equation in higher dimensions

We prove dispersive estimates at low frequency in dimensions n greater or equal to 4 for the wave equation for a very large class of real-valued potentials, provided the zero is neither an eigenvalue nor a resonance.

The Non-Linear Stochastic Wave Equation in High Dimensions

We propose an extension of Walsh's classical martingale measure stochastic integral that makes it possible to integrate a general class of Schwartz distributions, which contains the fundamental solution of the wave equation, even in dimensions greater than 3. This leads to a square-integrable random-field solution to the non-linear stochastic wave equation in any dimension, in the case of a driving noise that is white in time and correlated in space. In the particular case of an affine multiplicative noise, we obtain estimates on $p$-th moments of the solution ($p\geq 1$), and we show that the solution is Hölder continuous. The Hölder exponent that we obtain is optimal.

A selfadjoint second order hyperbolic system

We consider a second order hyperbolic system of the type (1) L u = u t t − B u x x = f ( x , t ) , ( x , t ) ∈ T m , where matrix B is a nonsingular constant matrix with positive eigenvalues, ( x , t ) ∈ R 2 and u , f ∈ R n . The set T m is defined to be (2) T m = { ( x , t ) | 0 ⩽ t ⩽ 1 / m , | x | ⩽ 1 − m t } , where m = min { μ k } and μ k 2 is any eigenvalue of the matrix B. We will show that, under the condition u ( x , 0 ) = 0 , | x | ⩽ 1 , a symmetric Green's function G n × n can be constructed [K. Kreith, A selfadjoint problem for the wave equation in higher dimensions, Comput. Math. Appl. 21 (5) (1991) 12–132] so that (3) u ( x , t ) = ∫ ∫ T m G n × n ( x , t ; ξ , τ ) f ( ξ , τ ) d ξ d τ for any function f ∈ L 2 ( T m ) . This will imply that the operator L in (1) over the set L 2 ( T m ) of functions given by Eq. (3) and u ( x , 0 ) = 0 , | x | ⩽ 1 , is selfadjoint. We also note that the same result holds for u in (1), under the condition that u t ( x , 0 ) = 0 , | x | ⩽ 1 . We further note that when B has only one eigenvalue μ 2 , the function u in Eq. (3) satisfies a boundary condition similar to that of Kalmenov [T. Kalmenov, On the spectrum of a selfadjoint problem for the wave equation, Akad. Nauk. Kazakh SSR Vestnik 1 (1983) 63–66] on the characteristic boundaries of T μ .