ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO THE HEAT EQUATIONS WITH NONLINEAR BOUNDARY CONDITIONS

The purpose of this paper is to investigate the stability and asymptotic behav-ior of the time-dependent solutions to a linear parabolic equation with nonlinear boundarycondition in relation to their corresponding steady state solutions. Then, the above resultsare extended to a semilinear parabolic equation with nonlinear boundary condition by an-alyzing the corresponding eigenvalue problem and using the method of upper and lowersolutions.     

Local properties of solutions of a semilinear elliptic equation with an inverse-square potential

We study the behavior at the origin of distribution solutions to a semilinear elliptic equation with an inverse-square potential. The relationship with nonuniqueness of solutions to the companion parabolic equation is discussed. __________ Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 17, Differential and Functional Differential Equations. Part 3, 2006.     

Space-time Estimates for Parabolic Type Operator and Application to Nonlinear Parabolic Equations

In this present paper we establish space-time estimates of solutions for linear parabolic type equations based on classical multipliers theory or operator semigroup theory. According to space-time estimates we first construct suitable work space L^q(0, T; L^P), moreover we study the Cauchy problem and initial boundary value problem for semilinear parabolic equation in L^q(0, T; L^P) type space.     

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.     

Existence of solutions for a higher-order semilinear parabolic equation with singular initial data

We establish the existence of solutions of the Cauchy problem for a higher-order semilinear parabolic equation by introducing a new majorizing kernel. We also study necessary conditions on the initial data for the existence of local-in-time solutions and identify the strongest singularity of the initial data for the solvability of the Cauchy problem.     

Sobolev multipliers,maximal functions and parabolic equations with a quadratic nonlinearity

We develop a general framework to describe global mild solutions to a Cauchy problem with small initial values concerning a general class of semilinear parabolic equations with a quadratic nonlinearity. This class includes the Navier–Stokes equations, the subcritical dissipative quasi-geostrophic equation and the parabolic–elliptic Keller–Segel system.     

Blow-up of solutions of a semilinear hyperbolic equation and a parabolic equation with general forcing term and boundary condition

Semilinear hyperbolic and parabolic initial–boundary value problems are studied. Criteria for solutions of a semilinear hyperbolic equation and a parabolic equation with general forcing term and general boundary condition to blow up in finite time are obtained.     

Finite time blow-up and global solutions for a class of semilinear parabolic equations at high energy level

In this paper we study the initial boundary value problem of a class of semilinear parabolic equation. Our main tools are the comparison principle and variational methods. In this paper, we will find both finite time blow-up and global solutions at high energy level.     

Slow convergence to zero for a parabolic equation with a supercritical nonlinearity

We study the behaviour of solutions of the Cauchy problem for a supercritical semilinear parabolic equation which converge to zero from above as t → ∞. We show that any algebraic decay rate slower than the self-similar one occurs for some initial data.     

Reflected BSDEs and the obstacle problem for semilinear PDEs in divergence form

We consider the Cauchy problem for a semilinear parabolic equation in divergence form with obstacle. We show that under natural conditions on the right-hand side of the equation and mild conditions on the obstacle, the problem has a unique solution and we provide its stochastic representation in terms of reflected backward stochastic differential equations. We also prove regularity properties and approximation results for solutions of the problem.     

Solutions with moving singularities for a semilinear parabolic equation

We consider the Cauchy problem for a semilinear heat equation with power nonlinearity. It is known that the equation has a singular steady state in some parameter range. Our concern is a solution with a moving singularity that is obtained by perturbing the singular steady state. By formal expansion, it turns out that the remainder term must satisfy a certain parabolic equation with inverse-square potential. From the well-posedness of this equation, we see that there appears a critical exponent. Paying attention to this exponent, for a prescribed motion of the singular point and suitable initial data, we establish the time-local existence, uniqueness and comparison principle for such singular solutions. We also consider solutions with multiple singularities.     

Harnack estimate for a semilinear parabolic equation

We study the Cauchy problem of a semilinear parabolic equation. We construct an appropriate Harnack quantity and get a differential Harnack inequality. Using this inequality, we prove the finite-time blow-up of the positive solutions and recover a classical Harnack inequality. We also obtain a result of Liouville type for the elliptic equation.     

Coefficient Stability of the Solution of a Difference Scheme Approximating a Mixed Problem for a Semilinear Parabolic Equation

We study the coefficient stability of a difference scheme approximating a mixed problem for a one-dimensional semilinear parabolic equation. We obtain sufficient conditions on the input data under which the solutions of the differential and difference problems are bounded. We also obtain estimates of perturbations of the solution of a linearized difference scheme with respect to perturbations of the coefficients; these estimates agree with the estimates for the differential problem.     

Propagation of Singularities of Solutions to Hyperbolic‐Parabolic Coupled Systems

In this paper, the authors study the propagation of singlarities for a semilinear hyperbolic‐parabolic coupled system, which comes from the model of thermoelasticity. Both of the Cauchy problem and the problem inside of a domain are considered. We obtain that the microlocal singularities of solutions to the semilinear hyperbolic‐parabolic coupled system are propagated along null bicharacteristics of the hyperbolic operator by using the theory of paradifferential operators. Furthermore, for the Cauchy problem of the semilinear coupled system, if the initial data have singularities at the origin, we prove that the solutions have the same order regularity with respect to spatial variables as in hyperbolic problems in the forward characteristic cone issuing from the origin, which improves the previous results for semilinear systems in thermoelasticity.     

Some blow-up problems for a semilinear parabolic equation with a potential

The blow-up rate estimate for the solution to a semilinear parabolic equation ut=Δu+V(x)|u|^p−1u in Ω×(0,T) with 0-Dirichlet boundary condition is obtained. As an application, it is shown that the asymptotic behavior of blow-up time and blow-up set of the problem with nonnegative initial data u(x,0)=Mφ(x) as M goes to infinity, which have been found in [C. Cortazar, M. Elgueta, J.D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, preprint, arXiv: math.AP/0607055, July 2006], is improved under some reasonable and weaker conditions compared with [C. Cortazar, M. Elgueta, J.D. Rossi, The blow-up problem for a semilinear parabolic equation with a potential, preprint, arXiv: math.AP/0607055, July 2006].     

IDENTIFICATION OF PARAMETERS IN SEMILINEAR PARABOLIC EQUATIONS

An optimization theoretic approach of coefficients in semilinear parabolic equation is presented. It is based on convex analysis techniques. General theorems on existence are proved in L¹ setting. A necessary condition is given for the solutions of the parameter estimation problem.     

On the structure of a solution set of controlled initial-boundary value problems

For a controlled nonlinear functional-operator equation of the Hammerstein type describing a wide class of controlled initial-boundary value problems, we obtain simple sufficient conditions for the convexity, pointwise boundedness and precompactness of the set of solutions (the reachability tube) in the Lebesgue space. As for boundedness and precompactness, we mean certain conditions of the majorant but not Volterra type requirements which give also the total (with respect to the whole set of admissible controls) preservation of solvability of mentioned equation. As some examples of reduction of a controlled initial-boundary (boundary) value problem to the equation under investigation, and verification of the proposed hypotheses for this equation, we consider the first initial-boundary value problem associated with a semilinear parabolic equation of the second order in a rather general form, and also the Dirichlet problem associated with a semilinear elliptic equation of the second order.     

关于猝灭问题的一些结果及其应用

本文研究一类含奇异项的半线性抛物方程的初边值问题,给出了该问题的古典解整体存在或发生猝灭现象的判据以及猝灭解的生命跨度估计。同时,还用上述结果研究了一类含超Sobolev临界指标的半线性椭圆方程的Dirichlet边值问题,获得了正解的存在性结果。     

A penalty approximation method for a semilinear parabolic double obstacle problem

In this work, we present a novel power penalty method for the approximation of a global solution to a double obstacle complementarity problem involving a semilinear parabolic differential operator and a bounded feasible solution set. We first rewrite the double obstacle complementarity problem as a double obstacle variational inequality problem. Then, we construct a semilinear parabolic partial differential equation (penalized equation) for approximating the variational inequality problem. We prove that the solution to the penalized equation converges to that of the variational inequality problem and obtain a convergence rate that is corresponding to the power used in the formulation of the penalized equation. Numerical results are presented to demonstrate the theoretical findings.     

Optimal lower bound of the grow-up rate for a supercritical parabolic equation

We study the behavior of solutions of the Cauchy problem for a supercritical semilinear parabolic equation which approach a singular steady state from below as t→∞. It is known that the grow-up rate of such solutions depends on the spatial decay rate of initial data. We give an optimal lower bound on the grow-up rate by using a comparison technique based on a formal asymptotic analysis.