On the finite-time singularities of the 3D incompressible Euler equations

We prove the finite-time vorticity blowup, in the pointwise sense, for solutions of the 3D incompressible Euler equations assuming some conditions on the initial data and its corresponding solutions near initial time. These conditions are represented by the relation between the deformation tensor and the Hessian of pressure, both coupled with the vorticity directions associated with the initial data and solutions near initial time. We also study the possibility of the enstrophy blowup for the 3D Euler and the 3D Navier-Stokes equations, and prove the finite-time enstrophy blowup for initial data satisfying suitable conditions. Finally, we obtain a new blowup criterion that controls the blowup by a quantity containing the Hessian of the pressure. © 2006 Wiley Periodicals, Inc.     

Spatial behaviour of solutions of the dual‐phase‐lag heat equation

In this paper we study the spatial behaviour of solutions of some problems for the dual‐phase‐lag heat equation on a semi‐infinite cylinder. The theory of dual‐phase‐lag heat conduction leads to a hyperbolic partial differential equation with a third derivative with respect to time. First, we investigate the spatial evolution of solutions of an initial boundary‐value problem with zero boundary conditions on the lateral surface of the cylinder. Under a boundedness restriction on the initial data, an energy estimate is obtained. An upper bound for the amplitude term in this estimate in terms of the initial and boundary data is also established. For the case of zero initial conditions, a more explicit estimate is obtained which shows that solutions decay exponentially along certain spatial‐time lines. A class of non‐standard problems is also considered for which the temperature and its first two time derivatives at a fixed time T are assumed proportional to their initial values. Copyright © 2004 John Wiley & Sons, Ltd.     

On a reaction diffusion equation with nonlinear time‐nonlocal source term

In this article, we prove the local existence of a unique solution to a nonlocal in time and space evolution equation with a time nonlocal nonlinearity of exponential growth. Moreover, under some suitable conditions on the initial data, it is shown that local solutions experience blow‐up. The time profile of the blowing‐up solutions is also presented. Copyright © 2015 John Wiley & Sons, Ltd.     

GLOBAL EXISTENCE OF SOLUTIONS FOR A MULTI-PHASE FLOW: A BUBBLE IN A LIQUID TUBE AND RELATED CASES

In this paper we study the problem of the global existence(in time) of weak,entropic solutions to a system of three hyperbolic conservation laws, in one space dimension,for large initial data. The system models the dynamics of phase transitions in an isothermal fluid; in Lagrangian coordinates, the phase interfaces are represented as stationary contact discontinuities. We focus on the persistence of solutions consisting in three bulk phases separated by two interfaces. Under some stability conditions on the phase configuration and by a suitable front tracking algorithm we show that, if the BV-norm of the initial data is less than an explicit(large) threshold, then the Cauchy problem has global solutions.     

On the entropy conditions for some flux limited diffusion equations

In this paper we give a characterization of the notion of entropy solutions of some flux limited diffusion equations for which we can prove that the solution is a function of bounded variation in space and time. This includes the case of the so-called relativistic heat equation and some generalizations. For them we prove that the jump set consists of fronts that propagate at the speed given by Rankine-Hugoniot condition and we give on it a geometric characterization of the entropy conditions. Since entropy solutions are functions of bounded variation in space once the initial condition is, to complete the program we study the time regularity of solutions of the relativistic heat equation under some conditions on the initial datum. An analogous result holds for some other related equations without additional assumptions on the initial condition.     

Spatial behaviour of solutions of the three-phase-lag heat equation

In this paper we study the spatial behaviour of solutions for the three-phase-lag heat equation on a semi-infinite cylinder. The theory of three-phase-lag heat conduction leads to a hyperbolic partial differential equation with a fourth-order derivative with respect to time. First, we investigate the spatial evolution of solutions of an initial boundary-value problem with zero boundary conditions on the lateral surface of the cylinder. Under a boundedness restriction on the initial data, an energy estimate is obtained. An upper bound for the amplitude term in this estimate in terms of the initial and boundary data is also established. For the case of zero initial conditions, a more explicit estimate is obtained which shows that solutions decay exponentially along certain spatial-time lines. A class of non-standard problems is also considered for which the temperature and its first two time derivatives at a fixed time T₀ are assumed proportional to their initial values. These results are relevant in the context of the Saint-Venant Principle for heat conduction problems.     

The Cauchy problem for the dissipative Boussinesq equation

This work is devoted to the solvability and finite time blow-up of solutions of the Cauchy problem for the dissipative Boussinesq equation in all space dimension. We prove the existence and uniqueness of local mild solutions in the phase space by means of the contraction mapping principle. By establishing the time-space estimates of the corresponding Green operators, we obtain the continuation principle. Under some restriction on the initial data, we further study the results on existence and uniqueness of global solutions and finite time blow-up of solutions with the initial energy at three different level. Moreover, the sufficient and necessary conditions of finite time blow-up of solutions are given.     

Regular and self-similar solutions of nonlinear Schrödinger equations

We study the Cauchy problem for the nonlinear Schrödinger equations with nonlinear term |u|^ou. For some admissible α we show the existence of global solutions and we calculate the regularity of those solutions. Also we give some necessary conditions and some sufficient conditions on initial data for the existence of self-similar solutions.     

Loss of boundary conditions for fully nonlinear parabolic equations with superquadratic gradient terms

We study whether the solutions of a fully nonlinear, uniformly parabolic equation with superquadratic growth in the gradient satisfy initial and homogeneous boundary conditions in the classical sense, a problem we refer to as the classical Dirichlet problem. Our main results are: the nonexistence of global-in-time solutions of this problem, depending on a specific largeness condition on the initial data, and the existence of local-in-time solutions for initial data $C^1$ up to the boundary. Global existence is know when boundary conditions are understood in the viscosity sense, what is known as the generalized Dirichlet problem. Therefore, our result implies loss of boundary conditions in finite time. Specifically, a solution satisfying homogeneous boundary conditions in the viscosity sense eventually becomes strictly positive at some point of the boundary.     

The self-similar solutions of the one-dimensional semilinear Tricomi-type equations

In this article we investigate the issue of existence of global in time solutions of semilinear Tricomi-type equations. We give conditions that relate the nonlinearity, the speed of propagation, and the order of singularities of initial data. These conditions guarantee existence of global in time solutions. In particular, we prove existence of solutions invariant under dilation by solving the Cauchy problem with initial data which are homogeneous functions.     

Remarks on asymptotic behaviors of strong solutions to a viscous Boussinesq system

In this paper, we first address the space‐time decay properties for higher‐order derivatives of strong solutions to the Boussinesq system in the usual Sobolev space. The decay rates obtained here are optimal. The proof is based on a parabolic interpolation inequality, bootstrap argument, and some weighted estimates. Secondly, we present a new solution integration formula for the Boussinesq system, which will be employed to establish the existence of strong solutions for small initial data in some scaling invariant function spaces. The smallness conditions are somehow weaker than those presented by Brandolese and Schonbek. We further investigate the asymptotic profiles and decay properties of these strong solutions. Copyright © 2016 John Wiley & Sons, Ltd.     

Existence and Asymptotic Behavior of the Wave Equation with Dynamic Boundary Conditions

The goal of this work is to study a model of the strongly damped wave equation with dynamic boundary conditions and nonlinear boundary/interior sources and nonlinear boundary/interior damping. First, applying the nonlinear semigroup theory, we show the existence and uniqueness of local in time solutions. In addition, we show that in the strongly damped case solutions gain additional regularity for positive times t>0. Second, we show that under some restrictions on the initial data and if the interior source dominates the interior damping term and if the boundary source dominates the boundary damping, then the solution grows as an exponential function. Moreover, in the absence of the strong damping term, we prove that the solution ceases to exists and blows up in finite time.     

Global Solvability and Estimates of Solutions to the Cauchy Problem for the Retarded Functional Differential Equations That Are Used to Model Living Systems

We study the Cauchy problem for the retarded functional differential equations that model the dynamics of some living systems. We find certain conditions ensuring the existence, uniqueness, and nonnegativity of solutions on finite and infinite time intervals. We obtain upper bounds for solutions and prove the continuous dependence of solutions on the initial data on finite time intervals.     

The pointwise estimates of solutions for a nonlinear convection diffusion reaction equation

This paper studies the time asymptotic behavior of solutions for a nonlinear convection diffusion reaction equation in one dimension.First,the pointwise estimates of solutions are obtained,furthermore,we obtain the optimal L~p,1≤ p ≤ +∞,convergence rate of solutions for small initial data.Then we establish the local existence of solutions,the blow up criterion and the sufficient condition to ensure the nonnegativity of solutions for large initial data.Our approach is based on the detailed analysis of the Green function of the linearized equation and some energy estimates.     

非线性Sobolev-Galpern方程解的blow up

本文研究非线性Ｓｏｂｏｌｅｖ－Ｃａｌｐｅｒｎ方程的初边值问题整体解的不存性即解的爆破问题,用能量估计方法并借助于Ｊｅｎｓｅｎ不等式证明了非线性Ｓｏｂｏｌｉｖ－Ｇａｌｐｅｒｎ方程各种初边值问题在某些假设下不存在整体解。     

Continuity with respect to initial data and absolute-continuity approach to the first-order regularity of nonlinear diffusions on noncompact manifolds

We study the dependence on initial data for solutions of diffusion equations with globally non-Lipschitz coefficients on noncompact manifolds. Though the metric distance may not be everywhere twice differentiable, we show that, under certain monotonicity conditions on the coefficients and curvature of the manifold, there are estimates exponential in time for the continuity of a diffusion process with respect to initial data. These estimates are combined with methods of the theory of absolutely continuous functions to achieve the first-order regularity of solutions with respect to initial data. The suggested approach neither appeals to the local stopping time arguments, nor applies the exponential mappings on the tangent space, nor uses imbeddings of a manifold to linear spaces of higher dimensions. Published in Ukrains’kyi Matematychnyi Zhurnal, Vol. 60, No. 10, pp. 1299–1316, October, 2008.     

End effects in three-phase-lag heat conduction

In this article we study the spatial behaviour of solutions for the three-phase-lag heat equation on a semi-infinite cylinder. First, we investigate the spatial evolution of solutions of an initial boundary-value problem with zero boundary conditions on the lateral surface of the cylinder. Under null initial data, a Phragmen–Lindelof alternative is obtained. An upper bound for the amplitude term in terms of the boundary data is also established. For the case of decay solutions, an improvement is obtained. We prove that the decay can be controlled by the exponential of a second-degree polynomial in the distance from the finite end of the cylinder. A class of non-standard problems is also considered for which the temperature and its first two time derivatives at a fixed time T ₀ are assumed proportional to their initial values. These results are relevant in the context of the Saint-Venant Principle for heat conduction problems.     

Structural stability of solutions to the Riemann problem for a scalar conservation law

The aim of this paper is to study the structural stability of solutions to the Riemann problem for a scalar conservation law with a linear flux function involving discontinuous coefficients. It is proved that the Riemann solution is possibly instable when one of the Riemann initial data is at the vacuum. Furthermore, we point out that the Riemann solution is also possibly instable even when the Riemann initial data stay far away from vacuum. In order to deal with it, we perturb the Riemann initial data by taking three piecewise constant states and then the global structures and large time asymptotic behaviors of the solutions are obtained constructively. It is also proved that the Riemann solutions are unstable in some certain situations under the local small perturbations of the Riemann initial data by letting the perturbed parameter ε tend to zero. In addition, the interaction of the delta standing wave and the contact vacuum state is considered which appear in the Riemann solutions.     

Blocking and invasion for reaction–diffusion equations in periodic media

We investigate the large time behavior of solutions of reaction–diffusion equations with general reaction terms in periodic media. We first derive some conditions which guarantee that solutions with compactly supported initial data invade the domain. In particular, we relate such solutions with front-like solutions such as pulsating traveling fronts. Next, we focus on the homogeneous bistable equation set in a domain with periodic holes, and specifically on the cases where fronts are not known to exist. We show how the geometry of the domain can block or allow invasion. We finally exhibit a periodic domain on which the propagation takes place in an asymmetric fashion, in the sense that the invasion occurs in a direction but is blocked in the opposite one.     

Large time existence of small viscous surface waves without surface tension

The motion of a three-dimensional viscous, imcompressible fluid is governed by the Navier-Stokes equations. We study the case where the fluid is in an ocean of infinite extent and finite depth with a free surface on top. This gives rise to a nonlinear free boundary problem. The given data are the initial velocity field and the initial free surface. In general, given smooth data, the solution will develop singularities in finite time; however, the effect of viscosity and surface tension tends to prevent the ingulitrities. It was previously known that when both are present, small, appropriately smooth solutions do not develop singularities; that is, smooth solutions exist globally in time. In this paper, we show that viscosity alone will prevent the formation of singularitics, even without surface tension; i.e., small smooth data which satisfy certain natural compatibility conditions, smooth solutions exist for all time. Uniqueness of the solution for any finite time interval is also proved.