Global solution of the Cauchy problem in nonlinear thermodi.usion in solid body

We prove a theorem about global existence (in time) of the solution to the initial‐value problem for a nonlinear hyperbolic parabolic system of coupled partial differential equation of second order describing the process of thermodiffusion in solid body. The corresponding global existence theorems has been proved using the L^p ‐ L^q time decay estimates for the solution of the associated linearized problem. Next, we proved the energy estimate in the Sobolev space with constant independent of time. Such an energy estimate allows us to apply the standard (continuation argument and to continue the local solution to one de.ned for all t ∈ 〈0, ∞)).     

Existence of global strong solution to the micropolar fluid system in a bounded domain

In this paper we are concerned with the initial boundary value problem of the micropolar fluid system in a three dimensional bounded domain. We study the resolvent problem of the linearized equations and prove the generation of analytic semigroup and its time decay estimates. In particular, L^p–L^q type estimates are obtained. By use of the L^p–L^q estimates for the semigroup, we prove the existence theorem of global in time solution to the original nonlinear problem for small initial data. Furthermore, we study the magneto‐micropolar fluid system in the final section. Copyright © 2005 John Wiley & Sons, Ltd.     

Lp – Lq time decay estimate to the solution of the Cauchy problem for the system of equations describing nonlocal model of the thermoviscoelastic body

The aim of this paper is to present a new system of equations describing nonlocal model of thermoviscoelastic theory. We used the Papkin and Gurtin approach based on the constitutive relations for stress tensor σ(x), internal energy e(x) and heat flux q(x), with integral terms. Using the modified Cagniard-de Hoop's method we constructed the matrix of fundamental solutions for this system of equations in three-dimensional space. Basing on this matrix we represent in the explicit formula the solution of the Cauchy problem to this system of equations. Next, applying the method of Sobolev spaces, we proved the L^p–L^q time decay estimate to the solution of the Cauchy problem. (© 2009 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

SUPERCRITICAL SEMILINEAR WAVE EQUATION WITH NON-NEGATIVE POTENTIAL

We prove a weighted L ^∞ estimate for the solution to the linear wave equation with a smooth positive time independent potential. The proof is based on application of generalized Fourier transform for the perturbed Laplace operator and a finite dependence domain argument. We apply this estimate to prove the existence of global small data solution to supercritical semilinear wave equations with potential.     

Global existence of solutions for one-dimensional compressible navier-stokes equations in the half space

We prove the existence of global solutions to the initial-boundary-value problem on the half space R＋ for a one-dimensional viscous ideal polytropic gas. Some suitable assumptions are made to guarantee the existence of smooth solutions. Employing the L2- energy estimate, we prove that the impermeable problem has a unique global solutionis.     

On the Global Solution of a 3‐D MHD System with Initial Data near Equilibrium

In this paper, we prove the global existence of smooth solutions to the three‐dimensional incompressible magnetohydrodynamical system with initial data close enough to the equilibrium state, (e₃,0). Compared with previous works by Lin, Xu, and Zhang and by Xu and Zhang, here we present a new Lagrangian formulation of the system, which is a damped wave equation and which is nondegenerate only in the direction of the initial magnetic field. Furthermore, we remove the admissible condition on the initial magnetic field, which was required in the earlier works. By using the Frobenius theorem and anisotropic Littlewood‐Paley theory for the Lagrangian formulation of the system, we achieve the global L¹‐in‐time Lipschitz estimate of the velocity field, which allows us to conclude the global existence of solutions to this system. In the case when the initial magnetic field is a constant vector, the large‐time decay rate of the solution is also obtained.© 2016 Wiley Periodicals, Inc.     

Qualitative analysis of a time‐delayed free boundary problem for tumor growth under the action of external inhibitors

In this paper, a time‐delayed free boundary problem for tumor growth under the action of external inhibitors is studied. It is assumed that the process of proliferation is delayed compared with apoptosis. By L^p theory of parabolic equations, the Banach fixed point theorem and the continuation theorem, the existence and uniqueness of a global solution is proved. The asymptotic behavior of the solution is also studied. The proof uses the comparison principle and the iteration method. Copyright © 2014 John Wiley & Sons, Ltd.     

L p – L q Estimates To The Solution Of The Cauchy Problem For Partial Differential Equations Describing Non-Simple Thermoelastic Materials

Theory of non-simple materials is different from that of simple materials because in it the first strain gradient is taken into consideration as the constitutive variable. The consequence of this fact, from mathematical point of view, is that the equation of motion consists either of higher order derivatives of displacement (four order derivatives) and some material parameters can depend not only on the temperature and the gradient of displacement but also on the second derivative of displacement. We consider the system of partial differential equations describing non-simple thermoelastic materials. This system consists of four scalar equations, three equations of motion and one of energy balance, describing the field of displacement and the temperature in an elastic body. Using the Fourier transform, we found the L ^p – L ^q time decay estimates of the solution of the Cauchy problem for the system of equation describing the non-simple thermoelastic materials, being important for proving the global-in-time solution of this problem. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Solvability of the Navier—Stokes System with L 2 Boundary Data

We prove the existence of the very weak solution of the Dirichlet problem for the Navier—Stokes system with L ² boundary data. Under the small data assumption we also prove the uniqueness. We use the penalization method to study the linearized problem and then apply Banach's fixed point theorem for the nonlinear problem with small boundary data. We extend our result to the case with no small data assumption by splitting the data on a large regular and small irregular part. Accepted 15 March 1999     

Nonexistence of Global Solutions to Nonlinear Stochastic Wave Equations in Mean L p -Norm

This article is concerned with explosive solutions of the initial-boundary problem for a class of nonlinear stochastic wave equations in a domain 𝒟 ? ? ^d . Under appropriate conditions on the initial data, the nonlinear term and the noise intensity, it is proved in Theorem 3.4 that there cannot exist a global solution and the local solution will blow up at a finite time in the mean L ^p  ? norm for p ≥ 1. An example is given to show the application of this theorem.     

On the initial value problem for the Vlasov-Poisson-Fokker-Planck system with initial data in Lp spaces

In this paper the global existence of weak solutions for the Vlasov-Poisson-Fokker-Planck equations in three dimensions is proved with an L¹ ∩ L^p initial data. Also, the global existence of weak solutions in four dimensions with small initial data is studied. A convergence of the solutions is obtained to those built by E. Horst and R. Hunze when the Fokker-Planck term vanishes. In order to obtain the a priori necessary estimates a sequence of approximate problems is introduced. This sequence is obtained starting from a non-linear regulation of the problem together with a linearization via a time retarded mollification of the non-linear term. The a priori bounds are reached by means of the control of the kinetic energy in the approximate sequence of problems. Then, the proof is completed obtaining the equicontinuity properties which allow to pass to the limit.     

The mean field kinetic equation for interacting particle systems with non-Lipschitz force

In this paper, we prove the global existence of the weak solution to the mean field kinetic equation derived from the N-particle Newtonian system. For L¹∩L^∞ initial data, the solvability of the mean field kinetic equation can be obtained by using uniform estimates and compactness arguments while the difficulties arising from the nonlocal nonlinear interaction are tackled appropriately using the Aubin-Lions compact embedding theorem.     

Sup-norm stability for Glimm's scheme

We consider the Cauchy problem for a general N × N system of conservation laws. Existence of solutions was proved by Glimm using his celebrated random choice scheme. In this paper, we obtain a third-order interaction estimate analagous to that obtained by Glimm for 2×2 systems. By using this estimate, and identifying a global cancellation effect, we obtain L^∞-stability for solutions generated by Glimm's scheme. As an immediate consequence we have L¹-stability and L^∞-decay, obtained by Temple for 2×2 systems. © 1993 John Wiley & Sons, Inc.     

Global strong solution to the three dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density‐dependent viscosity and resistivity

In this paper, we consider an initial boundary value problem for the 3‐dimensional nonhomogeneous incompressible magnetohydrodynamic equations with density‐dependent viscosity and resistivity coefficients over a bounded smooth domain. Global in time unique strong solution is proved to exist when the L² norms of initial vorticity and current density are both suitably small with arbitrary large initial density, and the vacuum of initial density is also allowed. Finally, we revisit the Navier‐Stokes model without electromagnetic effect. We find that this initial boundary problem also admits a unique global strong solution under other conditions. In particular, we prove small kinetic‐energy strong solution exists globally in time, which extends the recent result of Huang and Wang.     

Semi‐linear wave models with power non‐linearity and scale‐invariant time‐dependent mass and dissipation,II

This paper is a continuation of our recent paper 8 . We will consider the semi‐linear Cauchy problem for wave models with scale‐invariant time‐dependent mass and dissipation and power non‐linearity. The goal is to study the interplay between the coefficients of the mass and the dissipation term to prove global existence (in time) of small data energy solutions assuming suitable regularity on the L² scale with additional L¹ regularity for the data. In order to deal with this L² regularity in the non‐linear part, we will develop and employ some tools from Harmonic Analysis.     

On the global existence and uniqueness of solutions to Prandtl’s system

In this paper, we consider the Prandtl system for the non-stationary boundary layer in the vicinity of a point where the outer flow has zero velocity. It is assumed that U（t, x, y） = x^mU1（t, x）, where 0 〈 x 〈 L and m 〉 1. We establish the global existence of the weak solution to this problem. Moreover the uniqueness of the weak solution is proved.     

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.     

The global attractor of the viscous Fornberg–Whitham equation

This paper aims to present a proof of the existence of the attractor for the one-dimensional viscous Fornberg–Whitham equation. In this paper, the global existence of solution to the viscous Fornberg–Whitham equation in L² under the periodic boundary conditions is studied. By using the time estimate of the Fornberg–Whitham equation, we get the compact and bounded absorbing set and the existence of the global attractor for the viscous Fornberg–Whitham equation.     

Time decay of the solution to the Cauchy problem for a three-dimensional model of nonsimple thermoelasticity

This paper is devoted to the investigation of the solution to the Cauchy problem for a system of partial differential equations describing thermoelasticity of nonsimple materials in a three-dimensional space. The model of linear dynamical thermoelasticity of nonsimple materials is considered as the system of partial differential equations of fourth order. In this paper, we proposed a convenient evolutionary method of approach to the system of equations of nonsimple thermoelasticity. We proved the L^p−L^q time decay estimates for the solution to the Cauchy problem for linear thermoelasticity of nonsimple materials.     

Behaviors of solutions for a singular diffusion equation

This paper is devoted to some behaviors of solutions of the initial-boundary problem for a singular diffusion equation, namely, localization and large time behavior. After given some special explicit solutions it is proved that solutions of the problem possess the localization property. Next, L² decay estimate as t→∞ is proved by a rather standard energy method. Finally, by comparison with a special solution the expected L^∞ decay estimate is derived.