We prove estimates for solutions of the Cauchy problem for the inhomogeneous wave equation on $\R^{1+n}$ in a class of Banach spaces whose norms only depend on the size of the space-time Fourier transform. The estimates are local in time, and this allows one, essentially, to replace the symbol of the wave operator, which vanishes on the light cone in Fourier space, with an inhomogeneous symbol, which can be inverted. Our result improves earlier estimates of this type proved by Klainerman-Machedon. As a corollary, one obtains a rather general result concerning local well-posedness of nonlinear wave equations.
We consider a nonlinear parabolic equation with an exponential nonlinearity which is critical with respect to the growth of the nonlinearity and the regularity of the initial data. After showing the equivalence of the notions of weak and mild solutions, we derive decay estimates and the asymptotic behavior for small global-in-time solutions.
In this article, we consider a viscoelastic plate equation with past history, nonlinear damping, and logarithmic nonlinearity. We prove explicit and general decay rate results of the solution to the viscoelastic plate equation with past history. Convex properties, logarithmic inequalities, and generalised Young's inequality are mainly used to prove the decay estimate.
We establish Harnack's estimates for positive weak solutions to a mixed local and nonlocal doubly nonlinear parabolic equation. All results presented in this paper are provided together with quantitative estimates.
We derive a degenerate quasilinear Schrödinger equation that describes the resonant reflection of very weak, nonlinear sound waves off a weak sawtooth entropy wave.
In this short note, we present some decay estimates for nonlinear solutions of 3d quintic, 3d cubic and 2d quintic NLS (nonlinear Schrödinger equations).
We study the instability of standing waves for nonlinear Schrödinger equations. Under a general assumption on nonlinearity, we prove that linear instability implies orbital instability in any dimension. For that purpose, we establish a Strichartz type estimate for the propagator generated by the linearized operator around standing wave.
In this article, we consider a viscoelastic plate equation with a logarithmic nonlinearity in the presence of nonlinear frictional damping term. Using the the Faedo-Galerkin method we establish the global existence of the solution of the problem and we also prove few general decay rate results.
Hölder estimates and Harnack inequalities are studied for fully nonlinear integro-differential equations under some mild assumptions. We allow the kernels of variable order and critically close to 2.
In this paper, we establish a local gradient estimate for a $p$-Lpalacian equation with a fast growing gradient nonlinearity. With this estimate, we can prove a parabolic Liouville theorem for ancient solutions satisfying some growth restriction near infinity.
In this paper, the Lie symmetry method is performed for the nonlinear evolution equation for description nonlinear waves in a viscoelastic tube. we will find one and two-dimensional optimal system of Lie subalgebras. Furthermore, preliminary classification of its group-invariant solutions are investigated.
The well-known Stokes waves refer to periodic traveling waves under the gravity at the free surface of a two dimensional full water wave system. In this paper, we prove that small-amplitude Stokes waves with infinite depth are nonlinearly unstable under long-wave perturbations. Our approach is based on the modulational approximation of the water wave system and the instability mechanism of the focusing cubic nonlinear Schrödinger equation.
In this paper, we study the global existence and regularity of Hölder continuous solutions for a series of nonlinear partial differential equations describing nonlinear waves.
In this paper we study the {\it a priori} gradient estimates for admissible solutions to Neumann boundary value problem of fully nonlinear Hessian equations on Riemannian manifolds. We firstly derive an interior gradient estimates for admissible solutions, then we obtain boundary gradient estimates based on the interior gradient estimates we have got.
$W^{σ,p}$ estimates are studied for a class of fully nonlinear integro-differential equations of order $σ$, which are analogues of $W^{2,p}$ estimates by Caffarelli. We also present Aleksandrov-Bakelman-Pucci maximum principles, which are improvements of estimates proved by Guillen-Schwab, depending only on $L^p$ norms of inhomogeneous terms.
We consider nonlinear elastic wave equations generalizing Gol'dberg's five constants model. We analyze the nonlinear interaction of two distorted plane waves and characterize the possible nonlinear responses. Using the boundary measurements of the nonlinear responses, we solve the inverse problem of determining elastic parameters from the displacement-to-traction map.
We prove global Holder estimates for solution of fully nonlinear elliptic or degenerate elliptic equations in unbounded domains under geometric conditions on the domain a' la Cabre'.
For weak solutions to the evolutional $p$-Laplace equation with a time-dependent Radon measure on the right hand side we obtain pointwise estimates via a nonlinear parabolic potential.
We obtain Dini type estimates for a class of concave fully nonlinear nonlocal elliptic equations of order $σ\in (0,2)$ with rough and non-symmetric kernels. The proof is based on a novel application of Campanato's approach and a refined $C^{σ+α}$ estimate in [8].
We prove global well-posedness for the cubic, defocusing, nonlinear Schr{ö}dinger equation on $\mathbf{R}^{2}$ with data $u_{0} \in H^{s}(\mathbf{R}^{2})$, $s > 1/4$. We accomplish this by improving the almost Morawetz estimates in [9].
In this paper there are estimated the derivatives of the solution of an initial boundary value problem for a nonlinear uniformly parabolic equation in the interior with the total variation of the boundary data and the L^{infinity}-norm of the initial condition.
In this paper, we establish pointwise Schauder estimates for solutions of nonlocal fully nonlinear elliptic equations by perturbative arguments. A key ingredient is a recursive Evans-Krylov theorem for nonlocal fully nonlinear translation invariant equations.
We obtain the global weighted $W^{1,p}$ estimates for weak solutions of nonlinear elliptic equations over Reifenberg flat domains. Where nonlinearity $A(x,z,ξ)$ is assumed to be local uniform continuous in $z$ and have small BMO semi-norm in $x$. Moreover, we derive Besov regularity for solutions of a class of special harmonic equations by making use of $W^{1,p}$ estimate. Keywords: global weighted $W^{1,p}$ estimates; quasilinear equations; Besov regularity
The curvature estimates of quotient curvature equation do not always exist even for convex setting \cite{GRW}. Thus it is natural question to find other type of elliptic equations possessing curvature estimates. In this paper, we discuss the existence of curvature estimate for fully nonlinear elliptic equations defined by symmetric polynomials, mainlly, the linear combination of elementary symmetric polynomials.
Aim of this paper is to extend the continuous dependence estimates proved in \cite{JK1} to quasi-monotone systems of fully nonlinear second-order parabolic equations. As by-product of these estimates, we get an Hölder estimate for bounded solutions of systems and a rate of convergence estimate for the vanishing viscosity approximation. In the second part of the paper we employ similar techniques to study the periodic homogenization of quasi-monotone systems of fully nonlinear second-order uniformly parabolic equations. Finally, some examples are discussed.
We study the global existence and decay estimates for nonlinear wave equations with the space-time dependent dissipative term in an exterior domain. The linear dissipative effect may vanish in a compact space region. Moreover the nonlinear terms need not divergence form. For getting the higher order energy estimates, we introduce an argument using the rescaling. The method is useful to control derivatives of the dissipative coefficient.
We consider nonlinear Schrödinger equations with either power-type or Hartree nonlinearity in the presence of an external potential. We show that for long-range nonlinearities, solutions cannot exhibit scattering to solitary waves or more general localized waves. This extends the well-known results concerning non-existence of non-trivial scattering states for long-range nonlinearities.
The global attraction to stationary states is established for solutions to 3D wave equations with concentrated nonlinearities: each finite energy solution converges as $t\to\pm\infty$ to stationary states. The attraction is caused by nonlinear energy radiation.
Global smooth solutions to the initial value problem for systems of nonlinear wave equations with multiple propagation speeds will be constructed in the case of small initial data and nonlinearities satisfying the null condition.
For scalar semilinear wave equations, we analyze the interaction of two (distorted) plane waves at an interface between media of different nonlinear properties. We show that new waves are generated from the nonlinear interactions, which might be responsible for the observed nonlinear effects in applications. Also, we show that the incident waves and the nonlinear responses determine the location of the interface and some information of the nonlinear properties of the media. In particular, for the case of a jump discontinuity at the interface, we can determine the magnitude of the jump.
In this paper we prove the existence of finite traveling-wave type solutions to the nonlinear double degenerate parabolic equation of turbulent filtration with absorption.
In this paper, a nonlinear Schrödinger equation with an attractive (focusing) delta potential and a repulsive (defocusing) double power nonlinearity in one spatial dimension is considered. It is shown, via explicit construction, that both standing wave and equilibrium solutions do exist for certain parameter regimes. In addition, it is proved that both types of wave solutions are orbitally stable under the flow of the equation by minimizing the charge/energy functional.
This paper is concerned with the final value problem for a system of nonlinear wave equations. The main issue is to solve the problem for the case where the nonlinearity is of a long range type. By assuming that the solution is spherically symmetric, we shall show global solvability of the final value problem around a suitable final state, and hence the generalized wave operator and long range scattering operator can be constructed.
In this paper we consider a semilinear parabolic equation in an infinite cylinder. The spatially varying nonlinearity is such that it connects two (spatially independent) bistable nonlinearities in a compact set in space. We prove that, given such a setting, a traveling wave obeying the equation with the one bistable nonlinearity and starting at the respective side of the cylinder, will converge to a traveling wave solution prescribed by the nonlinearity on the other side.
We present a unified approach for obtaining global a priori estimates for solutions of nonlinear defocusing Schrödinger equations with defocusing nonlinearities. The estimates are produced by contracting the local momentum conservation law with appropriate vector fields. The corresponding law is written for defocusing equations of tensored solutions. In particular, we obtain a new estimate in two dimensions. We bound the restricted $L_t^4L_γ^4$ Strichartz norm of the solution on any curve $γ$ in $\mathbb R^2$. For the specific case of a straight line we upgrade this estimate to a weighted Stri…
We extend the notion of orbital stability to systems of nonlinear Schrodinger equations, then we prove this property under suitable assumptions of the local nonlinearity involved.
In this paper we introduce a finite-parameters feedback control algorithm for stabilizing solutions of various classes of damped nonlinear wave equations. Specifically, stabilization the zero steady state solution of initial boundary value problems for nonlinear weakly and strongly damped wave equations, nonlinear wave equation with nonlinear damping term and some related nonlinear wave equations, introducing a feedback control terms that employ parameters, such as, finitely many Fourier modes, finitely many volume elements and finitely many nodal observables and controllers. In addition, we a…
We examine $L^p$-viscosity solutions to fully nonlinear elliptic equations with bounded-measurable ingredients. By considering $p_0<p<d$, we focus on gradient-regularity estimates stemming from nonlinear potentials. We find conditions for local Lipschitz-continuity of the solutions and continuity of the gradient. We briefly survey recent breakthroughs in regularity theory arising from (nonlinear) potential estimates. Our findings follow from -- and are inspired by -- fundamental facts in the theory of $L^p$-viscosity solutions, and results in the work of Panagiota Daskalopoulos, Tuomo Kuusi an…
This article is concerned with the Schauder estimate for linear kinetic Fokker-Planck equations with Hölder continuous coefficients. This equation has an hypoelliptic structure. As an application of this Schauder estimate, we prove the global well-posedness of a toy nonlinear model in kinetic theory. This nonlinear model consists in a non-linear kinetic Fokker-Planck equation whose steady states are Maxwellian and whose diffusion in the velocity variable is proportional to the mass of the solution.
In this paper, we consider the nonlinear fractional Schrödinger equations with Hartree type nonlinearity. We obtain the existence of standing waves by studying the related constrained minimization problems by applying the concentration-compactness principle. Via symmetric decreasing rearrangements, we also show that the standing waves, up to a translations and phases, are positive symmetric nonincreasing functions. Moreover, we prove that the set of minimizers is a stable set for the initial value problem of the equations, that is, a solution whose initial data is near the set will remain near…
Consider a nonlinear wave equation for a massless scalar field with self-interaction in the spatially flat de Sitter spacetime. We show that blow-up in a finite time occurs for the equation with arbitrary power nonlinearity as well as upper bounds of the lifespan of blow-up solutions.The blow-up condition is the same as in the accelerated expanding Friedmann-Lemaître-Robertson-Walker (FLRW) spacetime. We also show the same results for the space derivative nonlinear term.
We obtain an error estimate between viscosity solutions and δ-viscosity solutions of nonhomogeneous fully nonlinear uniformly elliptic equations. The main assumption, besides uniform ellipticity, is that the nonlinearity is Lipschitz-continuous in space with linear growth in the Hessian. We also establish a rate of convergence for monotone and consistent finite difference approximation schemes for such equations.
In this article, we study the local existence of solutions for a wave equation with a nonlocal in time nonlinearity. Moreover, a blow-up results are proved under some conditions on the dimensional space, the initial data and the nonlinear forcing term.
This note establishes an interior quantitative lower bound for nonnegative supersolutions of fully nonlinear uniformly parabolic equations. The result may be interpreted as a nonlinear, quantitative version of a growth lemma established by Krylov and Safonov for nonnegative supersolutions of linear uniformly parabolic equations in nondivergence form. Our approach is different, and follows from an application of the Fabes-Stroock estimate. The result is the parabolic analogue of an elliptic regularity estimate established by Caffarelli, Souganidis, and Wang in the stochastic homogenization of f…
