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In this work, we investigate the global existence and asymptotic behavior of a mathematical model of nonlinear ultrasonic heating based on a coupled system of the Westervelt equation and the hyperbolic Pennes bioheat equation (Westervelt--Pennes--Cattaneo model). First, we prove that the solution exists globally in time, provided that the lower-order Sobolev norms of the initial data are considered to be small, while the higher-order norms can be arbitrarily large. This is done using a continuity argument together with some interpolation inequalities. Second, we prove an exponential decay of the solution under the same smallness assumptions on the initial data.

The evolution of an electrically conducting imcompressible fluid with nonconstant density can be described by a set of equations combining the continuity, momentum and Maxwell's equations; altogether known as the inhomogeneous Navier--Stokes--Maxwell system. In this paper, we focus on the global well-posedness of these equations in two dimensions. Specifically, we are able to prove the existence of global energy solutions, provided that the initial velocity field belongs to the Besov space $\dot{B}^{r}_{p,1}(\mathbb{R}^2)$, with $r=-1+\frac{2}{p}$, for some $p\in (1,2)$, while the initial electromagnetic field enjoys some $H^s(\mathbb{R}^2)$ Sobolev regularity, for some $s \geq 2-\frac{2}{p} \in (0,1)$, and whenever the initial fluid density is bounded pointwise and close to a nonnegative constant. Moreover, if it is assumed that $s>\frac{1}{2}$, then the solution is shown to be unique in the class of all energy solutions. It is to be emphasized that the solutions constructed here are global and uniformly bounded with respect to the speed of light $c\in (0,\infty)$. This important fact allows us to derive the inhomogeneous MHD system as the speed of light tends to infinity.

In this work, we investigate a mathematical model of nonlinear ultrasonic heating based on the Jordan-Moore-Gibson-Thompson equation (JMGT) with temperature-dependent medium parameters coupled to the semilinear Pennes equation for the bioheat transfer. The equations are coupled via the temperature in the coefficients of the JMGT equation and via a nonlinear source term within the Pennes equation, which models the absorption of acoustic energy by the surrounding tissue. Using the energy method together with a fixed point argument, we prove that our model is locally well-posed, provided that the initial data are regular, small in a lower topology and the final time is short enough., Comment: 37 pages

This paper aims to explore the long-term behavior of some nonlocal high-order-in-time wave equations. These equations, which have come to be known as Moore--Gibson--Thompson equations, arise in the context of acoustic wave propagation when taking into account thermal relaxation mechanisms in complex media such as human tissue. While the long-term behavior of linear local-in-time acoustic equations is well understood, their nonlocal counterparts still retain many mysteries. We establish here a set of assumptions that ensures exponential decay of the energy of the system. These assumptions are then shown to be verified by a large class of rapidly decaying memory kernels. Under weaker assumptions on the kernel we show that one may still obtain that the energy vanishes but without a rate of convergence. Furthermore, we refine previous results on the local well-posedness of the studied equation and establish a necessary initial-data compatibility condition for the solvability of the problem.

In this work, we investigate a mathematical model of nonlinear ultrasonic heating based on a coupled system of the Westervelt equation and the hyperbolic Pennes bioheat equation (Westervelt--Pennes--Cattaneo model). Using the energy method together with a fixed point argument, we prove that our model is locally well-posed and does not degenerate under a smallness assumption on the pressure data in the Westervelt equation. In addition, we perform a singular limit analysis and show that the Westervelt--Pennes--Fourier model can be seen as an approximation of the Westervelt--Pennes--Cattaneo model as the relaxation parameter tends to zero. This is done by deriving uniform bounds of the solution with respect to the relaxation parameter.

High frequencies at which ultrasonic waves travel give rise to nonlinear phenomena. In thermoviscous fluids, these are captured by Blackstock's acoustic wave equation with strong damping. We revisit in this work its well-posedness analysis. By exploiting the parabolic-like character of this equation due to strong dissipation, we construct a time-weighted energy framework for investigating its local solvability. In this manner, we obtain the small-data well-posedness on bounded domains under less restrictive regularity assumptions on the initial conditions compared to the known results. Furthermore, we prove that such initial boundary-value problems for the Blackstock equation are globally solvable and that their solution decays exponentially fast to the steady state.

High-Intensity Focused Ultrasound (HIFU) waves are known to induce localized heat to a targeted area during medical treatments. In turn, the rise in temperature influences their speed of propagation. This coupling affects the position of the focal region as well as the achieved pressure and temperature values. In this work, we investigate a mathematical model of nonlinear ultrasonic heating based on the Westervelt wave equation coupled to the Pennes bioheat equation that captures this so-called thermal lensing effect. We prove that this quasi-linear model is well-posed locally in time and does not degenerate under a smallness assumption on the pressure data.

In this paper, we consider the Cauchy problem of a model in nonlinear acoustic, named the Jordan--Moore--Gibson--Thompson equation. This equation arises as an alternative model to the well-known Kuznetsov equation in acoustics. We prove global existence and optimal time decay of solutions in Besov spaces with a minimal regularity assumption on the initial data, lowering the regularity assumption required in \cite{Racke_Said_2019} for the proof of the global existence. Using a time-weighted energy method with the help of appropriate Lyapunov-type estimates, we also extend the decay rate in \cite{Racke_Said_2019} and show an optimal decay rate of the solution for initial data in the Besov space $\dot{{B}}_{2,\infty}^{-3/2}(\mathbb{R}^3)$, which is larger than the Lebesgue space $L^1(\R^3)$ due to the embedding $L^1(\mathbb{R}% ^3)\hookrightarrow \dot{{B}}_{2,\infty}^{-3/2}(\mathbb{R}^3)$. Hence we removed the $L^1$-assumption on the initial data required in \cite{Racke_Said_2019} in order to prove the decay estimates of the solution.

In this paper, we consider the 3D Jordan--Moore--Gibson--Thompson equation arising in nonlinear acoustics. First, we prove that the solution exists globally in time provided that the lower order Sobolev norms of the initial data are considered to be small, while the higher-order norms can be arbitrarily large. This improves some available results in the literature. Second, we prove a new decay estimate for the linearized model and removing the $L^1$-assumption on the initial data. The proof of this decay estimate is based on the high-frequency and low-frequency decomposition of the solution together with an interpolation inequality related to Sobolev spaces with negative order.

Ultrasonic propagation through media with thermal and molecular relaxation can be modeled by third-order in time nonlinear wave-like equations with memory. This paper investigates the asymptotic behavior of a Cauchy problem for such a model, the nonlocal Jordan--Moore--Gibson--Thompson equation, in the so-called critical case, which corresponds to propagation in inviscid fluids. The memory has an exponentially fading character and type I, meaning that involves only the acoustic velocity potential. A major challenge in the global analysis is that the linearized equation's decay estimates are of regularity-loss type. As a result, the classical energy methods fail to work for the nonlinear problem. To overcome this difficulty, we construct appropriate time-weighted norms, where weights can have negative exponents. These problem-tailored norms create artificial damping terms that help control the nonlinearity and the loss of derivatives, and ultimately allow us to discover the model's asymptotic behavior.

In this paper, we prove the stability of Couette flow for 2D Navier-Stokes Boussinesq system without thermal diffusivity for the initial perturbation in Gevrey-$\frac{1}{s}$, ($1/3

We prove global solvability of the third-order in time Jordan-More-Gibson-Thompson acoustic wave equation with memory in $\mathbb{R}^n$, where $n \geq 3$. This wave equation models ultrasonic propagation in relaxing hereditary fluids and incorporates both local and cumulative nonlinear effects. The proof of global solvability is based on a sequence of high-order energy bounds that are uniform in time, and derived under the assumption of an exponentially decaying memory kernel and sufficiently small and regular initial data.

Motivated by the propagation of nonlinear sound waves through relaxing hereditary media, we study a nonlocal third-order Jordan-Moore-Gibson-Thompson acoustic wave equation. Under the assumption that the relaxation kernel decays exponentially, we prove local well-posedness in unbounded two- and three-dimensional domains. In addition, we show that the solution of the three-dimensional model exists globally in time for small and smooth data, while the energy of the system decays polynomially.

In this paper, we consider the standard linear solid model in $\mathbb{R}^N$ coupled, first, with the Fourier law of heat conduction and, second, with the Cattaneo law. First, we give the appropriate functional setting to prove the well-posedness of both models under certain assumptions on the parameters (that is, $0<\tau\leq \beta$). Second, using the energy method in the Fourier space, we obtain the optimal decay rate of a norm related to the solution both in the Fourier and the Cattaneo heat conduction models under the same assumptions on the parameters. More concretely, we prove that, when $0<\tau<\beta$, the model with heat conduction has the same decay rate as the Cauchy problem without heat conduction (see [Pellicer_Said-Houari_AMO_2017]) both under the Fourier and Cattaneo heat laws. Also, we are able to see that the difference between using a Fourier or Cattaneo law in the heat conduction is not in the decay rate, but in the fact that the Cattaneo coupling exhibits the well-know regularity loss phenomenon, that is, Cattaneo model requires a higher regularity of the initial data for the solution to decay. When $0<\tau=\beta$ (that is, when the dissipation comes through the heat conduction) we still have asymptotic stability in both heat coupling models, but with a slower decay rate. As we prove later, such stability is not possible in the absence of the heat conduction. We also prove the optimality of the previous decay rates for both models by using the eigenvalues expansion method. Finally, we complete the results in [Pellicer_Said-Houari_AMO_2017] by showing how the condition $0<\tau< \beta$ is not only sufficient but also necessary for the asymptotic stability of the problem without heat conduction.

In this paper, we consider nonlinear thermoelastic systems of Timoshenko type in a one-dimensional bounded domain. The system has two dissipative mechanisms being present in the equation for transverse displacement and rotation angle - a frictional damping and a dissipation through hyperbolic heat conduction modelled by Cattaneo's law, respectively. The global existence of small, smooth solutions and the exponential stability in linear and nonlinear cases are established., Comment: 23 pages

In this paper, we consider the Bresse system with frictional damping terms and prove some optimal decay results for the $L^2$-norm of the solution and its higher order derivatives. In fact, if we consider just one damping term acting on the second equation of the solution, we show that the solution does not decay at all. On the other hand, by considering one damping term alone acting on the third equation, we show that this damping term is strong enough to stabilize the whole system. In this case, we found a completely new stability number that depends on the parameters in the system. In addition, we prove the optimality of the results by using eigenvalues expansions. Our obtained results have been proved under some assumptions on the wave speeds of the three equations in the Bresse system., Comment: 27 pages

In this paper, we study the Moore--Gibson--Thompson equation in $\mathbb{R}^N$, which is a third order in time equation that arises in viscous thermally relaxing fluids and also in viscoelastic materials (then under the name of \emph{standard linear viscoelastic} model). First, we use some Lyapunov functionals in the Fourier space to show that, under certain assumptions on some parameters in the equation, an energy norm related with the solution decays with a rate $(1+t)^{-N/4}$. But this does not give the decay rate of the solution itself. Hence, in the second part of the paper, we show an explicit representation of the solution in the frequency domain by analyzing the eigenvalues of the Fourier image of the solution and writing the solution accordingly. We use this eigenvalues expansion method to give the decay rate of the solution (and its derivatives), which (for the solution) results in $(1+t)^{1-N/4}$ for $N=1,2$ and $(1+t)^{1/2-N/4}$ when $N\geq 3$.

The goal of this work is to study a model of the wave equation with dynamic boundary conditions and a viscoelastic term. First, applying the Faedo-Galerkin method combined with the fixed point theorem, we show the existence and uniqueness of a local in time solution. Second, we show that under some restrictions on the initial data, the solution continues to exist globally in time. On the other hand, if the interior source dominates the boundary damping, then the solution is unbounded and grows as an exponential function. In addition, in the absence of the strong damping, then the solution ceases to exist and blows up in finite time., Comment: arXiv admin note: text overlap with arXiv:0810.1013

This paper is concerned with decay estimate of solutions to the semilinear wave equation with strong damping in a bounded domain. Introducing an appropriate Lyaponuv function, we prove that when the damping is linear, we can find initial data, for which the solution decays exponentially. This result improves an early one in an article of Gazzola and Squassina.

In this paper we consider a multi-dimensional wave equation with dynamic boundary conditions, related to the Kelvin-Voigt damping. Global existence and asymptotic stability of solutions starting in a stable set are proved. Blow up for solutions of the problem with linear dynamic boundary conditions with initial data in the unstable set is also obtained.

In this paper we consider a multi-dimensional damped semiliear wave equation with dynamic boundary conditions, related to the Kelvin-Voigt damping. We firstly prove the local existence by using the Faedo-Galerkin approximations combined with a contraction mapping theorem. Secondly, the exponential growth of the energy and the $L^p$ norm of the solution is presented.