The genuinely nonlinear Graetz—Nusselt ultraparabolic equation

We study a second-order quasilinear ultraparabolic equation whose matrix of the coefficients of the second derivatives is nonnegative, depends on the time and spatial variables, and can change rank in the case when it is diagonal and the coefficients of the first derivatives can be discontinuous. We prove that if the equation is a priori known to enjoy the maximum principle and satisfies the additional “genuine nonlinearity” condition then the Cauchy problem with arbitrary bounded initial data has at least one entropy solution and every uniformly bounded set of entropy solutions is relatively compact in L _loc ¹ . The proofs are based on introduction and systematic study of the kinetic formulation of the equation in question and application of the modification of the Tartar H-measures proposed by E. Yu. Panov.     

Threshold phenomena for symmetric decreasing solutions of reaction-diffusion equations

We study the long time behavior of solutions of the Cauchy problem for nonlinear reaction-diffusion equations in one space dimension with the nonlinearity of bistable, ignition or monostable type. We prove a one-to-one relation between the long time behavior of the solution and the limit value of its energy for symmetric decreasing initial data in L ² under minimal assumptions on the nonlinearities. The obtained relation allows to establish sharp threshold results between propagation and extinction for monotone families of initial data in the considered general setting.     

Regularity of the Interfaces with Sign Changes of Solutions of the One-Dimensional Porous Medium Equation

In previous papers we considered the Cauchy problem for the one-dimensional evolution p-Laplacian equation for nonzero, bounded, and nonnegative initial data having compact support, and showed that after a finite time the set of spatial critical points of the nonnegative solution u=u(x, t) in {u>0} consists of one point, the spatial maximum point of u, and the curve of the spatial maximum points is continuous with respect to the time variable. Since the spatial derivative ∂_xu satisfies the porous medium equation with sign changes, the curve of the spatial maximum points is regarded as an interface with sign changes of ∂_xu. On the other hand, in a paper by M. Bertsch and D. Hilhorst (1991, Appl. Anal.41, 111-130) the interfaces where the solutions change their sign were studied in detail for the initial-boundary value problems of the generalized porous medium equation over two-dimensional cylinders. But the monotonicity of the initial data is assumed there. As is noted in Section 4 of our earlier work (1996, J. Math. Anal. Appl.203, 78-103), the monotonicity of ∂_xu(?, t) in some neighborhood of the spatial maximum point of u(?, t) cannot be assumed, and therefore, if this monotonicity for some large t>0 is proved, then by the method of Bertsch and Hilhorst (cited above) one may get more precise regularity properties of the curve of the spatial maximum points. The purpose of the present paper is twofold. One is to remove some monotonicity assumption for initial data in Bertsch and Hilhorst's theorem concerning the regularity of the interfaces with sign changes of solutions of the one-dimensional generalized porous medium equation. By comparing the solution with appropriate symmetric nonnegative solutions we shall get the monotonicity of the solution near the interface after a finite time. The other is as a by-product of the method to get C¹ regularity of the curves of the spatial maximum points of nonnegative solutions of the Cauchy problem for the evolution p-Laplacian equation for sufficiently large t.     

On the ratio of different norms of caloric functions and related eigenfunctions of an integral operator

We investigate the ratio of L 1 and L 2 norms of the Cauchy problem solutions of heat equations with compact support initial data.The related asymptotic behavior of the eigenvalues and eigenfunctions of certain integral operators is obtained.     

Parametric resonance of ground states in the nonlinear Schrödinger equation

We study the global existence and long-time behavior of solutions of the initial-value problem for the cubic nonlinear Schrödinger equation with an attractive localized potential and a time-dependent nonlinearity coefficient. For small initial data, we show under some nondegeneracy assumptions that the solution approaches the profile of the ground state and decays in time like t^-1/4. The decay is due to resonant coupling between the ground state and the radiation field induced by the time-dependent nonlinearity coefficient.     

Pointwise estimates for the fundamental solutions of a class of singular parabolic problems

We deal with the Cauchy problem associated to a class of quasilinear singular parabolic equations with L ^∞ coefficients whose prototypes are the p-Laplacian (2N/(N + 1) < p < 2) and the porous medium equation (((N ? 2)/N)₊ < m < 1). We prove existence of and sharp pointwise estimates from above and from below for the fundamental solutions. Our results can be extended to general non-negative L ¹ initial data.     

Global solutions of the evolutionary Faddeev model with small initial data

We consider the Cauchy problem for evolutionary Faddeev model corresponding to maps from the Minkowski space ℝ¹⁺ⁿ to the unit sphere $ \mathbb{S} $ \mathbb{S} ², which obey a system of non-linear wave equations. The nonlinearity enjoys the null structure and contains semi-linear terms, quasi-linear terms and unknowns themselves. We prove that the Cauchy problem is globally well-posed for sufficiently small initial data in Sobolev space.     

Global existence for systems of parabolic conservation laws in several space variables

We prove the global existence of solutions of the Cauchy problem for certain systems of conservation laws with artificial viscosity terms added. The system is assumed to admit a quadratic entropy which is consistent with the viscosity matrix, and the initial data is assumed to be close to a constant in L² ∩ L^∞. In particular, our result applies to the equations of compressible fluid flow in two and three space variables.     

Global analysis of smooth solutions to a hyperbolic-parabolic coupled system

We investigate a model arising from biology, which is a hyperbolic- parabolic coupled system. First, we prove the global existence and asymptotic behavior of smooth solutions to the Cauchy problem without any smallness assumption on the initial data. Second, if the Hs ∩ Ll-norm of initial data is sufficiently small, we also establish decay rates of the global smooth solutions. In particular, the optimal L2 decay rate of the solution and the almost optimal L2 decay rate of the first-order derivatives of the solution are obtained. These results are obtained by constructing a new nonnegative convex entropy and combining spectral analysis with energy methods.     

Stability of nonlinear Schrödinger equations on modulation spaces

We consider the Cauchy problem for a family of SchrSdinger equations with initial data in modulation spaces Mp,1^s. We develop the existence, uniqueness, blowup criterion, stability of regularity, scattering theory, and stability theory.     

Existence of Infinite Energy Solution to the Inelastic Boltzmann Equation with External Force

In this paper, the Cauchy problem for the inelastic Boltzmann equation with external force is considered in the case of initial data with infinite energy. More precisely, under the assumptions on the bicharacteristic generated by external force, we prove the global existence of solution for small initial data compared to the local Maxwellian exp{–p|x – v|²}, which has infinite mass and energy.     

Asymptotics of odd solutions for cubic nonlinear Schrödinger equations

We consider the Cauchy problem for a cubic nonlinear Schrödinger equation in the case of an odd initial data from H²∩H0,2. We prove the global existence in time of solutions to the Cauchy problem and construct the modified asymptotics for large values of time.     

Asymptotics of odd solutions of quadratic nonlinear Schrödinger equations

We consider the Cauchy problem for a quadratic nonlinear Schrödinger equation in the case of odd initial data from H²∩H0,2. We prove the global existence in time of solutions to the Cauchy problem and construct the modified asymptotics for large values of time.     

Further regularity results for almost cubic nonlinear Schrödinger equation

We present three results related with the regularity of solutions of the almost cubic NLS. In the first one, following Ozawa’s idea, we establish mass and energy conservation for the solutions without regularizing the initial datum. Our second result is the H^s well-posedness for the Cauchy problem for 0<s<1. Finally, we show that the same solutions are also in some Bourgain spaces for possibly a smaller time interval. In all of our results, the non-local nonlinear term in the equation is shown to act like a cubic nonlinearity on the appropriate Sobolev and Besov spaces.     

Porous medium equations on manifolds with critical negative curvature: unbounded initial data

We investigate existence and uniqueness of solutions of the Cauchy problem for the porous medium equation on a class of Cartan–Hadamard manifolds. We suppose that the radial Ricci curvature, which is everywhere nonpositive as well as sectional curvatures, can diverge negatively at infinity with an at most quadratic rate: in this sense it is referred to as critical. The main novelty with respect to previous results is that, under such hypotheses, we are able to deal with unbounded initial data and solutions. Moreover, by requiring a matching bound from above on sectional curvatures, we can also prove a blow-up theorem in a suitable weighted space, for initial data that grow sufficiently fast at infinity.     

Optimal convergence rates to nonlinear diffusion waves for the compressible Euler-Poisson system with damping

In this work, we study the 1-D isentropic bipolar hydrodynamic model. This model takes the form of compressible Euler-Poisson system with nonlinear damping added to the momentum equations. Under some smallness conditions, the solutions to the Cauchy problem of the system globally exist and convergence to the nonlinear diffusion waves, which are the corresponding solutions of nonlinear parabolic equations given by the Darcy's law with a specified initial data. The optimal convergence rates are obtained by Green function method when the initial perturbation is in L¹-space.     

Entropy solutions for linearly degenerate hyperbolic systems of rich type

Consider a linearly degenerate hyperbolic system of rich type. Assuming that each eigenvalue of the system has a constant multiplicity, we construct a representation formula of entropy solutions in L^∞ to the Cauchy problem. This formula depends on the solution of an autonomous system of ordinary differential equations taking x as parameter. We prove that for smooth initial data, the Cauchy problem for such an autonomous system admits a unique global solution. By using this formula together with classical compactness arguments, we give a very simple proof on the global existence of entropy solutions. Moreover, in a particular case of the system, we obtain an another explicit expression and the uniqueness of the entropy solution. Applications include the one-dimensional Born–Infeld system and linear Lagrangian systems.     

Grow-up rate of solutions for a supercritical semilinear diffusion equation

We study solutions of the Cauchy problem for a supercritical semilinear parabolic equation which converge to a singular steady state from below as t→∞. We show that the grow-up rate of such solutions depends on the spatial decay rate of initial data.     

Estimates for a solution to one differential inequality

In a domain D = Ω × (?T,T) we consider a differential inequality whose left-hand side contains a linear second-order hyperbolic operator with coefficients depending only on x ∈ ? ⁿ, n ≥ 2, and the right-hand side contains the modulus of the gradient of the sought function. We supplement the inequality with the Cauchy data on the lateral part of the boundary of D and consider the problem of estimating a solution to the differential inequality satisfying the Cauchy data. We establish the estimate under some assumptions that involves the upper bound of the sectional curvatures of the Riemannian space associated with the differential operator, the Riemannian diameter of Ω, and the length of the interval (?T,T). The result is generalized to the case of compact domains bounded from above and below by characteristic surfaces.     

Scattering for focusing combined power-type NLS

We consider the scattering of Cauchy problem for the focusing combined power-type Schr¨odinger equation. In the spirit of concentration-compactness method, we will show that, H1 solution will scatter under some condition on its energy and mass. We adapt some variance argument, following the idea of Ibrahim–Masmoudi–Nakanishi.