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In this work, we investigate a class of quasilinear wave equations of Westervelt type with, in general, nonlocal-in-time dissipation. They arise as models of nonlinear sound propagation through complex media with anomalous diffusion of Gurtin–Pipkin type. Aiming at minimal assumptions on the involved memory kernels – which we allow to be weakly singular – we prove the well-posedness of such wave equations in a general theoretical framework. In particular, the Abel fractional kernels, as well as Mittag-Leffler-type kernels, are covered by our results. The analysis is carried out uniformly with respect to the small involved parameter on which the kernels depend and which can be physically interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the behavior of solutions as this parameter vanishes, and in this way relate the equations to their limiting counterparts. To establish the limiting problems, we distinguish among different classes of kernels and analyze and discuss all ensuing cases.

In ultrasonics, nonlocal quasilinear wave equations arise when taking into account a class of heat flux laws of Gurtin–Pipkin type within the system of governing equations of sound motion. The present study extends previous work by the authors to incorporate nonlocal acoustic wave equations with quadratic gradient nonlinearities which require a new approach in the energy analysis. More precisely, we investigate the Kuznetsov and Blackstock equations with dissipation of fractional type and identify a minimal set of assumptions on the memory kernel needed for each equation. In particular, we discuss the physically relevant examples of Abel and Mittag–Leffler kernels. We perform the well-posedness analysis uniformly with respect to a small parameter on which the kernels depend and which can be interpreted as the sound diffusivity or the thermal relaxation time. We then analyze the limiting behavior of solutions with respect to this parameter, and how it is influenced by the specific class of memory kernels at hand. Through such a limiting study, we relate the considered nonlocal quasilinear equations to their limiting counterparts and establish the convergence rates of the respective solutions in the energy norm. [ABSTRACT FROM AUTHOR]

We prove Strichartz estimates for solutions to Maxwell equations in three dimensions with rough permittivities, which have less than three different eigenvalues. To this end, Maxwell equations are conjugated to half-wave equations in phase space. We use the Strichartz estimates in a known combination with energy estimates to show the new well-posedness results for quasilinear Maxwell equations. [ABSTRACT FROM AUTHOR]

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 $ 1.1-1.4 $ are new. Moreover, the changing trends of semilinear wave equations are illustrated through numerical simulation. [ABSTRACT FROM AUTHOR]

In this paper, one-stage explicit trigonometric integrators for solving quasilinear wave equations are formulated and studied. For solving wave equations, we first introduce trigonometric integrators as the semidiscretization in time and then consider a spectral Galerkin method for the discretization in space. We show that one-stage explicit trigonometric integrators in time have second-order convergence and the result is also true for the fully discrete scheme without requiring any CFL-type coupling of the discretization parameters. The results are proved by using energy techniques, which are widely applied in the numerical analysis of methods for partial differential equations. [ABSTRACT FROM AUTHOR]

We consider an inverse problem arising in nonlinear ultrasound imaging. The propagation of ultrasound waves is modeled by a quasilinear wave equation. We make measurements at the boundary of the medium encoded in the Dirichlet-to-Neumann map and we show that these measurements determine the nonlinearity. [ABSTRACT FROM AUTHOR]

We are concerned with the initial-boundary value problems of 3-D quasilinear wave equations outside compact convex obstacles with Neumann boundary conditions. When the surfaces of 3-D compact convex obstacles are smooth and the quadratic nonlinearities in the quasilinear wave equations do not fulfill the null condition, we establish the almost global existence of smooth small-amplitude solutions to the above initial-boundary value problems. The lower bound of the lifespan is proved to be eκ/ε with some positive κ and the small positive parameter ε as the size of the initial data. This lower bound is optimal as shown in the 3-D boundaryless case. [ABSTRACT FROM AUTHOR]

In this paper, the initial-boundary value problems of quasilinear wave equations with acoustic boundary conditions, nonlinear internal and boundary source term was considered. When the initial energy is assumed to be negative, the global nonexistence of the solution of the quasilinear wave equation is proved by constructing Lyapunov functional. [ABSTRACT FROM AUTHOR]

This work deals with the convergence analysis of parabolic perturbations to quasilinear wave equations on smooth bounded domains. In particular, we consider wave equations with nonlinearities of quadratic type, which cover the two classical models of nonlinear acoustics, the Westervelt and Kuznetsov equations. By employing a high-order energy analysis, we obtain convergence of their solutions to the corresponding inviscid equations' solutions in the standard energy norm with a linear rate, assuming small data and a sufficiently short time. The smallness of initial data can, however, be imposed in a lower-order norm than the one needed in the energy analysis. It arises only from ensuring the nondegeneracy of the studied wave equations. In addition, we address the open questions of local well-posedness of the two classical models on bounded domains with a nonnegative, possibly vanishing sound diffusivity. [ABSTRACT FROM AUTHOR]

We consider the Kirchhoff equation ∂tt u -- △u (1 + ∫Td | ▽u|²) = 0 on the d-dimensional torus Td, and its Cauchy problem with initial data u(0, x), ∂t u(0, x) of size ε in the Sobolev class. The effective equation for the dynamics at the quintic order, obtained in previous papers by quasilinear normal form, contains resonances corresponding to nontrivial terms in the energy estimates. Such resonances cannot be avoided by tuning external parameters (simply because the Kirchhoff equation does not contain parameters). In this paper we introduce nonresonance conditions on the initial data of the Cauchy problem and prove a lower bound ε-6 for the lifespan of the corresponding solutions (the standard local theory gives ε-2, and the normal form for the cubic terms gives ε-4). The proof relies on the fact that, under these nonresonance conditions, the growth rate of the "superactions" of the effective equations on large time intervals is smaller (by a factor ε²) than its a priori estimate based on the normal form for the cubic terms. The set of initial data satisfying such nonresonance conditions contains several nontrivial examples that are discussed in the paper. [ABSTRACT FROM AUTHOR]

Consider the Kirchhoff equation ∂ tt u - Δ u (1 + ∫ T d | ∇ u | 2) = 0 on the d-dimensional torus T d . In a previous paper we proved that, after a first step of quasilinear normal form, the resonant cubic terms show an integrable behavior, namely they give no contribution to the energy estimates. This leads to the question whether the same structure also emerges at the next steps of normal form. In this paper, we perform the second step and give a negative answer to the previous question: the quintic resonant terms give a nonzero contribution to the energy estimates. This is not only a formal calculation, as we prove that the normal form transformation is bounded between Sobolev spaces. [ABSTRACT FROM AUTHOR]

In this paper, we study the Cauchy problem for systems of 3-D quasilinear wave equations satisfying the null condition with initial data of low regularity. In the radially symmetric case, we prove the global existence for every small data in H 3 × H 2 with a low weight. To achieve this goal, we will show how to extend the global iteration method first suggested by Li and Chen (1988) [32] to the low regularity case, which is also another purpose of this paper. Finally, we apply our result to 3-D nonlinear elastic waves. [ABSTRACT FROM AUTHOR]

In this article, we prove the almost global existence of solutions for quasilinear wave equations in the complement of star-shaped domains in three dimensions, with a Neumann boundary condition.

In this paper, we give an alternative proof to the global existence result, which is originally owing to the pioneering work of Klainerman and Christodoulou, for the Cauchy problem of quasilinear wave equations with null condition in three space dimensions. The proof can display the following three features simultaneously: the Lorentz boost operator is not employed in the generalized energy estimates; the generalized energy of the solution will be always small, which was first observed by Alinhac; and the initial data are not assumed to have a compact support. Copyright © 2016 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We give an alternative proof of Alinhac's almost global existence result for the Cauchy problem of quasilinear wave equations with quadratic nonlinearity satisfying the null condition in 2-D (S. Alinhac, 2001, [3] ). The main innovation of our proof is that the Lorentz boost operator is not employed in the generalized energy estimates. [ABSTRACT FROM AUTHOR]

We study the initial–boundary value problem of quasilinear wave equations outside of a star-shaped obstacle in four space dimensions, in which the nonlinear term under consideration may explicitly depend on the unknown function itself. By some new L t ∞ L x 2 and weighted L t , x 2 estimates for the unknown function itself, together with energy estimates and KSS type estimates, for the quasilinear obstacle problem we obtain a lower bound of the lifespan T ε ≥ exp ⁡ ( c ε 2 ) , which coincides with the sharp lower bound of lifespan estimate for the corresponding Cauchy problem. [ABSTRACT FROM AUTHOR]

Based on the theory of semi-global piecewise C2 solutions to 1D quasilinear wave equations, the local exact boundary controllability of nodal profile for quasilinear wave equations in a planar tree-like network of strings with general topology is obtained by a constructive method. The principles of providing nodal profiles and of choosing and transferring boundary controls are presented, respectively. Copyright © 2013 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]

We study the Cauchy problem for systems of quasilinear wave equations with multiple propagation speeds in four space dimensions. The nonlinear term in this problem may explicitly depends on the unknown function itself. By some new Lt∞ Lx². estimates of the unknown function, combining with some Klainerman--Sideris type weighted estimates, we get the sharp lower bound of lifespan Tε ≥ exp (c/ε²) for the quasilinear system. [ABSTRACT FROM AUTHOR]

We study the minimal surface equation in Minkowskian geometry in , which is a well-known quasilinear wave equation. The classical result of Lindblad, [[Proc. Amer. Math. Soc. 132: 1095--1102, 2004]], establishes global existence of small and smooth solutions (i.e. global regularity), provided the initial data is small, compactly supported and very smooth. In the present paper, we achieve more precise results. We show that, at least when n ≥≥ 4 (or n == 3, but with radial data), it is enough to assume the smallness of some scale invariant quantities, involving (unweighted) Sobolev norms only. In the 3D case, such a proof fails as a consequence of the well-known Strichartz inequality ''missing endpoint'' and one has instead slightly weaker results, which requires smallness of the data in certain Ws,p, p < 2, spaces. In the 2D case, this fails as well, since the free solution of the 2D wave equation fails to be square integrable, but only belongs to L2,∞∞, another failure by an endpoint. Thus, an important question is left open: Can one prove global regularity for the 2D minimal surface equation, assuming smallness in unweighted Sobolev spaces only? [ABSTRACT FROM AUTHOR]

We consider smooth solutions of radial exterior Cauchy–Neumann problems with small initial data for radial quasilinear wave equations in three space dimensions, when the size of the initial data tends to 0. We obtain rather precise information on the lifespan, analogous with well known Cauchy problem results of Hörmander and John. [ABSTRACT FROM AUTHOR]

Abstract: This work investigates the existence of globally Lipschitz continuous solutions to a class of initial–boundary value problem of quasilinear wave equations. Applying the Lax''s method and generalized Glimm''s method, we construct the approximate solutions of initial–boundary Riemann problem near the boundary layer and perturbed Riemann problem away from the boundary layer. By showing the weak convergence of residuals for the approximate solutions, we establish the global existence for the derivatives of solutions and obtain the existence of global Lipschitz continuous solutions of the problem. [Copyright &y& Elsevier]

Abstract: This work investigates the existence of globally Lipschitz continuous solutions to a class of Cauchy problem of quasilinear wave equations. Applying Lax''s method and generalized Glimm''s method, we construct the approximate solutions of the corresponding perturbed Riemann problem and establish the global existence for the derivatives of solutions. Then, the existence of global Lipschitz continuous solutions can be carried out by showing the weak convergence of residuals for the source term of equation. [Copyright &y& Elsevier]

Abstract: This paper presents an existence proof for symmetric modulating pulse solutions of a quasilinear wave equation. Modulating pulse solutions consist of a pulse-like envelope advancing in the laboratory frame and modulating an underlying wave train; they are also referred to as ‘moving breathers’ since they are time periodic in a moving frame of reference. The problem is formulated as an infinite-dimensional dynamical system with two stable, two unstable and infinitely many neutral directions. Using a partial normal form and a generalisation of local invariant-manifold theory to the quasilinear setting we prove the existence of modulating pulses on arbitrarily large, but finite domains in space and time. [Copyright &y& Elsevier]

Abstract: We consider the generalized quasilinear dissipative Kirchhoff''s String problemwith the initial conditions and , in the case where . The purpose of our work is to study the stability of the initial solution for this equation using central manifold theory. [Copyright &y& Elsevier]

The formation of singularity of solutions to the quasilinear wave equations with scattering damping and divergence form nonlinearities is investigated. The upper bound lifespan estimates of solutions to the problems with variable coefficient in the critical case and constant coefficient in the sub-critical case are established by applying the rescaled test function method and iterative method. [ABSTRACT FROM AUTHOR]