The stabilizability of heat equations with time-dependent coefficients

In this paper, we study the stabilizability of heat equations with time-dependent coefficients. The instability of the systems is illustrated by special solutions. The boundary feedback stabilization results are established by the backstepping method where the growth order of the kernel function depends on time.     

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.     

Regularity of Pressure in the Neighbourhood of Regular Points of Weak Solutions of the Navier-Stokes Equations

In the context of the weak solutions of the Navier-Stokes equations we study the regularity of the pressure and its derivatives in the space-time neighbourhood of regular points. We present some global and local conditions under which the regularity is further improved.     

Combining MFS and PGD methods to solve transient heat equation

We propose in this article a numerical algorithm based on the combination of the method of fundamental solutions (MFS) and the proper generalized decomposition technique (PGD) to solve time‐dependent heat equation. The MFS is considered as a truly meshless technique well adapted for a wide range of physical problems and the PGD approach can be considered as a reduction technique based on the separated representation of the variable functions. The proposed study relates to a separation between the spatial and temporal coordinates. To show the effectiveness of the proposed algorithm, several examples are presented and compared to the reference results.     

Numerical solution of the heat equation with nonlinear boundary conditions in unbounded domains

The numerical solution of the heat equation in unbounded domains (for a 1D problem‐semi‐infinite line and for a 2D one semi‐infinite strip) is considered. The artificial boundaries are introduced and the exact artificial boundary conditions are derived. The original problems are transformed into problems on finite domains. The space semi‐discretization by finite element method and the full approximation by the implicit‐explicit Euler's method are presented. The solvability of the full discretization schemes is analyzed. Computational examples demonstrate the accuracy and the efficiency of the algorithms. Also, the behavior of blowing up solutions is examined numerically. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 23: 379–399, 2007     

On the existence and uniqueness of time periodic solutions for a semilinear heat equation in the whole space

This paper is concerned with the existence and uniqueness of time periodic solutions in the whole‐space for a heat equation with nonlinear term. The nonlinear term we considered is of this type, |u |^{q ? 1}u + f (x ,t ), with , N > 2. We show that there exists a unique time periodic solution when the source term f is small. In fact, is a critical exponent; when , there is no time periodic solution. Copyright © 2016 John Wiley & Sons, Ltd.     

On a nonlinear heat equation with viscoelastic term associated with Robin conditions

This paper is devoted to study of a nonlinear heat equation with a viscoelastic term associated with Robin conditions. At first, by the Faedo–Galerkin and the compactness method, we prove existence, uniqueness, and regularity of a weak solution. Next, we prove that any weak solution with negative initial energy will blow up in finite time. Finally, by the construction of a suitable Lyapunov functional, we give a sufficient condition to guarantee the global existence and exponential decay of weak solutions.     

The Hardy inequality and the heat equation with magnetic field in any dimension

In the Euclidean space of any dimension d, we consider the heat semigroup generated by the magnetic Schrödinger operator from which an inverse-square potential is subtracted to make the operator critical in the magnetic-free case. Assuming that the magnetic field is compactly supported, we show that the polynomial large-time behavior of the heat semigroup is determined by the eigenvalue problem for a magnetic Schrödinger operator on the (d ? 1)-dimensional sphere whose vector potential reflects the behavior of the magnetic field at the space infinity. From the spectral problem on the sphere, we deduce that in d = 2 there is an improvement of the decay rate of the heat semigroup by a polynomial factor with power proportional to the distance of the total magnetic flux to the discrete set of flux quanta, while there is no extra polynomial decay rate in higher dimensions. To prove the results, we establish new magnetic Hardy-type inequalities for the Schrödinger operator and develop the method of self-similar variables and weighted Sobolev spaces for the associated heat equation.     

Symmetry of integral equation systems on bounded domains

In this paper, we investigate the symmetry of domains and solutions of integral equation systems on bounded domains. Under some natural integrability conditions, we prove that the domains are balls, all positive solutions of systems are radially symmetric and monotone decreasing with respect to the radius.     

Structure of the sets of regular and singular radial solutions for a semilinear elliptic equation

This paper is concerned with the structure of the set of radially symmetric solutions for the equation

     

Asymptotic behavior for the filtration equation in domains with noncompact boundary

We consider the initial value boundary problem with zero Neumann data for an equation modeled after the porous media equation, with variable coefficients. The spatial domain is unbounded and shaped like a (general) paraboloid, and the solution u is integrable in space and nonnegative. We show that the asymptotic profile for large times of u is one dimensional and given by an explicit function, which can be regarded as the fundamental solution of a one-dimensional differential equation with weights. In the case when the domain is a cone or the whole space (Cauchy problem), we obtain a genuine multidimensional profile given by the well-known Barenblatt solution.     

Numerical studies of a nonlinear heat equation with square root reaction term

Interest in calculating numerical solutions of a highly nonlinear parabolic partial differential equation with fractional power diffusion and dissipative terms motivated our investigation of a heat equation having a square root nonlinear reaction term. The original equation occurs in the study of plasma behavior in fusion physics. We begin by examining the numerical behavior of the ordinary differential equation obtained by dropping the diffusion term. The results from this simpler case are then used to construct nonstandard finite difference schemes for the partial differential equation. A variety of numerical results are obtained and analyzed, along with a comparison to the numerics of both standard and several nonstandard schemes. © 2008 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2009     

The asymptotic behavior of solutions of the sine-gordon equation with singularities at zero

The problem of asymptotic analysis of radially symmetric solutions of the sine-Gordon equation reducible to the third Painlevé transcendent is posed. Solutions with singularities at the origin are studied. For finite values of the independent variable, an asymptotic expansion of such a solution is obtained; the leading term of this expansion is a modulated elliptic function. The corresponding modulation equation and phase shift are written out. Translated fromMatematicheskie Zametki, Vol. 67, No. 3, pp. 329–342, March, 2000.     

Exact controllability of the heat equation with bilinear control

This paper deals with exact controllability of bilinear heat equation. Namely, given the initial state, we would like to provide a class of target states that can be achieved through the heat equation at a finite time by applying multiplicative controls. For this end, an explicit control strategy is constructed. Simulations are provided. Copyright © 2015 John Wiley & Sons, Ltd.     

The structure of solutions of a hypergeometric equation system

We consider a system of ordinary differential equations which is a multidimensional analogue of a hypergeometric equation. We study the structure and asymptotics of solutions at the singular points and construct a fundamental system of solutions in a neighborhood of each singular point. Institute of Mathematics and Informatics, Akademijos 4, 2600 Vilnius, Lithuania. Translated from Lietuvos Matematikos Rinkinys, Vol. 39, No. 1, pp. 52–64, January–March, 1999. Translated by V. Mackevičius     

Asymptotics of solutions to the time-dependent Schrödinger equation with a small Planck constant

A regularized asymptotics of the solution to the time-dependent Schrödinger equation in which the spatial derivative is multiplied by a small Planck constant is constructed. It is shown that the asymptotics of the solution contains a rapidly oscillating boundary layer function.     

Numerical solution of the one‐dimensional heat equation on the bounded intervals using fundamental solutions

We present a numerical method for the solution of heat equation with sufficiently smooth initial condition, using fundamental solutions of heat equation in terms of singularities. In this work various aspects of this method such as efficiency, stability, and convergency are given and a comparison with some well‐known finite difference methods will be obtained. Numerical results are reported to support the superiority of the developed method. © 2007 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2008     

Solutions with time-dependent singular sets for the heat equation with absorption

We consider the heat equation with a superlinear absorption term ${?}_{t}u?\mathrm{\Delta }u=?u^p$ in ${\mathbb{R}}^n$ and study the existence of nonnegative solutions with an m-dimensional time-dependent singular set, where $n?m\ge 3$. We prove that if $1<p<(n?m)/(n?m?2)$, then there are two types of singular solutions. Moreover, we show the uniqueness of the solutions and specify the exact behavior of the solutions near the singular set.     

Rough hypoellipticity for the heat equation in Dirichlet spaces

This paper aims at proving the local boundedness and continuity of solutions of the heat equation in the context of Dirichlet spaces under some rather weak additional assumptions. We consider symmetric local regular Dirichlet forms, which satisfy mild assumptions concerning (1) the existence of cut-off functions, (2) a local ultracontractivity hypothesis, and (3) a weak off-diagonal upper bound. In this setting, local weak solutions of the heat equation, and their time derivatives, are shown to be locally bounded; they are further locally continuous, if the semigroup admits a locally continuous density function. Applications of the results are provided including discussions on the existence of locally bounded heat kernel; $L^{\infty }$ structure results for ancient (local weak) solutions of the heat equation.     

On the structure of global solutions of the nonlinear heat equation in a ball

In this paper, we consider the nonlinear heat equation

(NLH)

u_t−Δu=|u|^αu,