Energy estimates for wave equations with time dependent propagation speeds in the Gevrey class

The total energy of the wave equation is conserved with respect to time if the propagation speed is a constant, but this is not true in general for time dependent propagation speeds. Indeed, it is considered in Hirosawa (2007) [3] that the following properties of the propagation speed are crucial for the estimates of the total energy: oscillating speed, difference from the mean, and the smoothness in Cm category. The main purpose of this paper is to derive a benefit of a further smoothness of the propagation speed in the Gevrey class for the energy estimates.     

On the asymptotic behavior of the energy for the wave equations with time depending coefficients

We consider the asymptotic behavior of the total energy of solutions to the Cauchy problem for wave equations with time dependent propagation speed. The main purpose of this paper is that the asymptotic behavior of the total energy is dominated by the following properties of the coefficient: order of the differentiability, behavior of the derivatives as t → ∞ and stabilization of the amplitude described by an integral. Moreover, the optimality of these properties are ensured by actual examples. Supported by Grants-in-Aid for Young Scientists (B) (No.16740098), The Ministry of Education, Culture, Sports, Science and Technology.     

Cauchy–Dirichlet problem for second-order hyperbolic equations in cylinders with non-smooth base

This paper is concerned with the smoothness of generalized solutions of the Cauchy–Dirichlet problem for the second-order hyperbolic equation in domains with a conical point.     

Secular growth estimates for hyperbolic systems

We consider a model problem for the secular growth, which covers all the cases likely to happen in multi-scales BKW expansions for nonlinear hyperbolic systems. This model problem consists in studying the growth in time of the solution of a nonhomogeneous hyperbolic system whose source term is a product of various functions which solve homogeneous hyperbolic systems. The secular growth is due to resonances, that we try to control. When this is not possible, other tools such as decay properties or Strichartz estimates must be used.     

Quasianalytic and nonquasianalytic solutions for a class of weakly hyperbolic Cauchy problems

In the weakly hyperbolic Cauchy problem, we investigate the relation between the modulus of continuity in the time variable of the coefficients and the well-posedness in Beurling-Roumieu classes of ultradifferentiable functions and functionals. We find a sharp condition on the modulus of continuity assuring the well-posedness in nonquasianalytic classes.     

A well-posedness result for hyperbolic operators with Zygmund coefficients

In this paper we prove an energy estimate with no loss of derivatives for a strictly hyperbolic operator with Zygmund continuous second order coefficients both in time and in space. In particular, this estimate implies the well-posedness for the related Cauchy problem. On the one hand, this result is quite surprising, because it allows to consider coefficients which are not Lipschitz continuous in time. On the other hand, it holds true only in the very special case of initial data in H^1/2×H^−1/2$H^{1/2}\times H^{-1/2}$. Paradifferential calculus with parameters is the main ingredient to the proof.     

Averaging principle and hyperbolic evolution equations

An averaging principle is derived for the abstract nonlinear evolution equation where the almost periodic right hand-side is a continuous perturbation of the time-dependent family of linear operators determining a linear evolution system. It generalizes classical Henry’s results for perturbations of sectorial operators on fractional spaces. It is also proved that the main hypothesis of the nonlinear averaging principle is satisfied for general hyperbolic evolution equations introduced by Kato.     

Asymptotic profiles for wave equations with strong damping

We consider the Cauchy problem in Rⁿ${\mathbf{R}}^n$ for strongly damped wave equations. We derive asymptotic profiles of these solutions with weighted L^1,1(Rⁿ)$L^{1,1}({\mathbf{R}}^n)$ data by using a method introduced in [9] and/or [10].     

Hardy-Sobolev extremals, hyperbolic symmetry and scalar curvature equations

We prove nondegeneracy of extremals for some Hardy-Sobolev-Maz'ya inequalities and present applications to scalar curvature-type problems, including the Webster scalar curvature equation in a cylindrically symmetric setting. The main theme is hyperbolic symmetry.     

Blow-up theorem for semilinear wave equations with non-zero initial position

One of the features of solutions of semilinear wave equations can be found in blow-up results for non-compactly supported data. In spite of finite propagation speed of the linear wave, we have no global in time solution for any power nonlinearity if the spatial decay of the initial data is weak. This was first observed by Asakura (1986) [2] finding out a critical decay to ensure the global existence of the solution. But the blow-up result is available only for zero initial position having positive speed.In this paper the blow-up theorem for non-zero initial position by Uesaka (2009) [22] is extended to higher-dimensional case. And the assumption on the nonlinear term is relaxed to include an example, |u|^p−1u. Moreover the critical decay of the initial position is clarified by example.     

Existence, non-existence and asymptotic behavior of global solutions to the Cauchy problem for systems of semilinear hyperbolic equations with damping terms

We consider the Cauchy problem for systems of semilinear hyperbolic equations. Using the L_p→L_q type estimation for the corresponding linear parts, the existence and uniqueness of weak global solutions are investigated. We also established the behavior of solutions and their derivatives as t→+∞. Using the method of test functions developed in the works (Mitidieri and Pokhozhaev, 2001 [11], Veron and Pohozaev, 2001 [12] and Caristi, 2000 [23]) we obtain the analogue of the Fujita-Hayakawa type criterion for the absence of global solutions to some system of semilinear hyperbolic inequalities with damping. It follows that the conditions of existence theorem imposed on the growth of nonlinear parts are exact in some sense.     

Decay estimates for quasi-linear evolution equations

We consider global strong solutions of the quasi-linear evolution equations (1.1) and (1.2) below, corresponding to sufficiently small initial data, and prove some stability estimates, as t→+∞, that generalize the corresponding estimates in the linear case.     

Sharp decay estimates for hyperbolic balance laws

We consider strictly hyperbolic and genuinely nonlinear systems of hyperbolic balance laws in one-space dimension. Sharp decay estimates are derived for the positive waves in an entropy weak solution. The result is obtained by introducing a partial ordering within the family of positive Radon measures, using symmetric rearrangements and a comparison with a solution of Burgers's equation with impulsive sources as well as lower semicontinuity properties of continuous Glimm-type functionals.     

Weighted estimates for stationary Navier-Stokes equations

We show Morrey-type estimates for the weak solution of the periodic Navier-Stokes equations in dimensionN, 5 <N < 10. ForN < 8, we prove the existence of a maximum solution.     

Decay estimates for a thermoelastic system in waveguides

We investigate the linear system of thermoelasticity, consisting of an elasticity equation and a heat conduction equation, in a waveguide Ω=(0,1)×Rn−1, with certain boundary conditions. We consider the cases of homogeneous and inhomogeneous systems and prove decay estimates of the solutions, which are a key ingredient to showing the global existence of solutions to non-linear thermoelasticity, after having decomposed the solutions into various parts. We also give a simplified proof to the representation of the solutions to the Cauchy problem of thermoelasticity.     

Energy decay rate for a coupled hyperbolic system with nonlinear damping

In this paper we investigate the energy decay rate for the solution of a coupled hyperbolic system. The explicit energy decay rate is established by using multiplier techniques and constructing a suitable energy functional.     

Global weighted estimates for quasilinear elliptic equations with non-standard growth

In this paper we obtain a new global gradient estimates in weighted Lorentz spaces for weak solutions of p(x)$p(x)$-Laplacian type equation with small BMO coefficients in a δ-Reifenberg flat domain. The modified Vitali covering lemma, the maximal function technique and the appropriate localization method are the main analytical tools. Our results improve the known results for such equations.     

Parabolic equations with BMO coefficients in Lipschitz domains

In this paper, we are concerned with certain natural Sobolev-type estimates for weak solutions of inhomogeneous problems for second-order parabolic equations in divergence form. The geometric setting is that of time-independent cylinders having a space intersection assumed to be locally given by graphs with small Lipschitz coefficients, the constants of the operator being uniformly parabolic. We prove the relevant L^p estimates, assuming that the coefficients are in parabolic bounded mean oscillation (BMO) and that their parabolic BMO semi-norms are small enough.