On the Fujita exponent for a semilinear heat equation with a potential term

We consider the existence and nonexistence of positive global solutions for the Cauchy problem,

     

On the determination of unknown source in the heat equation with nonlocal boundary conditions

We consider the inverse problem for the heat equation with unknown source. The existence and uniqueness conditions are established in the case where the boundary and overdetermination conditions are nonlocal conditions of general form.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 47, No. 10, pp. 1438–1441, October, 1995.     

Borel summability of the heat equation with variable coefficients

We prove Borel summability in nonsingular directions of solutions of partial differential equations of the form $u_t=a(z)u_{zz}$ where $a(z)$ is a quartic polynomial and the initial condition is analytic. In the special case $a(z)=z$ we obtain the detailed resurgent structure; even in such a simple case, the structure of the singular manifolds is quite intricate.     

On critical Fujita exponents for heat equations with nonlinear flux conditions on the boundary

We consider nonnegative solutions of initial-boundary value problems for parabolic equationsu _t=u_xx, u_t=(u^m)_xxand (m>1) forx>0,t>0 with nonlinear boundary conditions−u _x=u^p,−(u ^m)_x=u^pand forx=0,t>0, wherep>0. The initial function is assumed to be bounded, smooth and to have, in the latter two cases, compact support. We prove that for each problem there exist positive critical valuesp ₀,p_c(withp ₀<p_c)such that forp∃(0,p ₀],all solutions are global while forp∃(p₀,p_c] any solutionu≢0 blows up in a finite time and forp>p _csmall data solutions exist globally in time while large data solutions are nonglobal. We havep _c=2,p _c=m+1 andp _c=2m for each problem, whilep ₀=1,p ₀=1/2(m+1) andp ₀=2m/(m+1) respectively. This work was done during visits of the first author to Iowa State University and the Institute for Mathematics and its Applications at the University of Minnesota. The second author was supported in part by NSF Grant DMS-9102210.     

The heat equation with stochastic non-linear boundary conditions

In this paper we examine the problem of the heat equation with non-linear boundary conditions of stochastic type.     

Analytic transient solutions of a cylindrical heat equation with a heat source

The method of separation of variables is applied in order to investigate the analytical solutions of a certain two-dimensional cylindrical heat equation. In the analysis presented here, the partial differential equation is directly transformed into ordinary differential equations. The closed-form transient temperature distributions and heat transfer rates are generalized for a linear combination of the products of Fourier-Bessel series of the exponential type. Relevant connections with some other closely-related recent works are also indicated.     

Investigation of a nonlinear heat equation with a quadratic source

We consider a particular case of the nonlinear heat equation on a straight line. A family of exact solutions of the form p(t) + q(t) cos (x/ ) is constructed, where p(t) and q(t) satisfy some dynamical system. A detailed analysis of the system is given. The existence of blowup solutions as well as solutions that decay to a nonzero background is proved for the Cauchy problem for the given equation. Part of the solutions from this family are close in a certain sense to the analytical solution of the nonlinear equation with power nonlinearities evolving in the S-regime. Profiles of various solutions are constructed and localization is investigated numerically. __________ Translated from Prikladnaya Matematika i Informatika, No. 24, pp. 5–23, 2006.     

The heat equation with nonlinear generalized Robin boundary conditions

Let Ω⊂RN be a bounded domain and let μ be an admissible measure on ∂Ω. We show in the first part that if Ω has the H¹-extension property, then a realization of the Laplace operator with generalized nonlinear Robin boundary conditions, formally given by on ∂Ω, generates a strongly continuous nonlinear submarkovian semigroup SB=(SB(t))_t?0 on L²(Ω). We also obtain that this semigroup is ultracontractive in the sense that for every u,v∈Lp(Ω), p?2 and every t>0, one has

     

The nonlinear heat equation with absorption: Effects of variable coefficients

We consider the nonnegative solutions to the nonlinear degenerate parabolic equation u_t = (D(x, t)u^{m − 1}u_x)_x − b(x, t)u^p with m > 1, 0 < p < 1, and positive D(x, t), b(x, t). After obtaining the uniqueness and Hölder regularity results, we investigate the dependence of such phenomena as extinction in finite time and instantaneous shrinking of the support on the behaviour of D(x, t) and b(x, t).     

Remarks on a semilinear heat equation with integral boundary conditions

For a semilinear heat equation we consider a nonlocal boundary problem. On the basis of the solution of a Dirichlet problem for a parabolic equation and Volterra integral equation we establish the well-posedness for the nonlocal problem, which generalizes some recent results.     

Meshless and analytical solutions to the time-dependent advection-diffusion-reaction equation with variable coefficients and boundary conditions

A variety of physical problems in science may be expressed using the advection-diffusion-reaction (ADR) equation that covers heat transfer and transport of mass and chemicals into a porous or a nonporous media. In this paper, the meshless generalised reproducing kernel particle method (RKPM) is utilised to numerically solve the time-dependent ADR problem in a general n-dimensional space with variable coefficients and boundary conditions. A time-dependent Robin boundary condition is formulated and precisely enforced in a novel approach. The accuracy and robustness of the meshless solution is verified against finite element simulations and a general one-dimensional analytical solution obtained in this study.     

Heat equation with dynamical boundary conditions of reactive-diffusive type

This paper deals with the heat equation posed in a bounded regular domain Ω of RN (N?2) coupled with a dynamical boundary condition of reactive-diffusive type. In particular we study the problem

     

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.     

Exact solutions to mKdV equation with variable coefficients

In this paper a special mKdV with variable coefficients is considered. A transformation of variables is first applied in order to obtain a mKdV equation with constant coefficients. Some its one-, two- and three-soliton as well as breather-type soliton solutions are derived by using Hirorta’s bilinear approach.     

带非局部边界条件的热方程的LEGENDRE配置法

对带非局部边界条件的热方程的初边值问题提出了LEGENDRE配置法,并给出其半离散逼近和全离散逼近的稳定性和收敛性分析.数值试验验证了方法的有效性.     

An initial-boundary value problem with mixed lateral conditions for heat equation

Sunto Viene studiato un problema con condizioni laterali miste per l'equazione del calore, nel caso in cui la ? superficie di separazione ? abbia un punto caratteristico. Viene dato un teorema di esistenza e di unicità in spazi con peso a regolarità variabile.

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Entrata in Redazione l'8 marzo 1978.

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Partially supported by Istituto Nazionale di Fisica Nucleare, Sezione di Bologna, Bologna, Italy.