We consider a non-local filtration equation of the form

and a porous medium equation, in this case K(u) = um, with some boundary and initial data u0, where 0 < p < 1 and f, f′, f″ > 0. We prove blow-up of solutions for sufficiently large values of the parameter λ > 0 and for any u0 > 0, or for sufficiently large values of u0 > 0 and for any λ λ 0.

The freezing of water to ice is a classic problem in applied mathematics, involving the solution of a diffusion equation with a moving boundary. However, when the water is salty, the transport of salt rejected by ice introduces some interesting twists to the tale. A number of analytic models for the freezing of water are briefly reviewed, ranging from the famous work by Neumann and Stefan in the 1800s, to the mushy zone models coming out of Cambridge and Oxford since the 1980s. The successes and limitations of these models, and remaining modelling issues, are considered in the case of freezing sea-water in the Arctic and Antarctic Oceans. A new, simple model which includes turbulent transport of heat and salt between ice and ocean is introduced and solved analytically, in two different cases—one where turbulence is given by a constant friction velocity, and the other where turbulence is buoyancy-driven and hence depends on ice thickness. Salt is found to play an important role, lowering interface temperatures, increasing oceanic heat flux, and slowing ice growth.

We discuss a Conley index calculation which is of importance in population models with large interaction. In particular, we prove that a certain Conley index is trivial.

This paper is mainly concerned with the critical extinction and blow-up exponents for the homogeneous Dirichlet boundary-value problem of the fast diffusive polytropic filtration equation with reaction sources.

In this paper an ill-posed problem for the heat equation is investigated. Solutions u to the equation ut – uxx = 0, which are approximately known on the positive half-axis t = 0 and on some vertical lines x = x1,…, x = xn, are considered and stability estimates of these solutions are presented. We assume an a priori bound, governing the heat flow across the boundary x = 0.

Let ui(x, t) be the ith component of a nonnegative solution of a weakly coupled system of second-order, linear, parabolic partial differential equations, for x ∈ Rn and 0 < t < T. We obtain lower estimates, near t = 0, for the Lebesgue measure of the set of x for which exceeds 1. Related results for Poisson integrals on a half-space are also described, some applications are given, and interesting comparisons emerge.

We study singularities of solutions of the heat equation, that are not necessarily isolated but occur only in a single characteristic hyperplane. We prove a decomposition theorem for certain solutions on D+ = D ∩ (Rn × ]0. ∞[), for a suitable open set D, with singularities at compact subset K of Rn × {0}, in terms of Gauss-Weierstrass integrals. We use this to prove a representation theorem for certain solutions on D+, with singularities at K, as the sums of potentials and Dirichlet solutions. We also give conditions under which K is removable for solutions on D∖K.

We prove a priori estimates for the gradient and curvature of spacelike hypersurfaces moving by mean curvature in a Lorentzian manifold. These estimates are obtained under much weaker conditions than have been previously assumed. We also use mean curvature flow in the construction of maximal slices in asymptotically flat spacetimes. An essential tool is a maximum principle for sub-solutions of a parabolic operator on complete Riemannian manifolds with time-dependent metric.

We discuss expansions of solutions of the generalized heat equation which have a singularity at zero in terms of two sequences of homogeneous solutions.

We prove two asymptotical estimates for minimizers of a Ginzburg-Landau functional of the form .

In this paper we extend a theorem of Mallet-Paret and Sell for the existence of an inertial manifold for a scalar-valued reaction diffusion equation to new physical domains ωn ⊂ Rn, n = 2,3. For their result the Principle of Spatial Averaging (PSA), which certain domains may possess, plays a key role for the existence of an inertial manifold. Instead of the PSA, we define a weaker PSA and prove that the domains φn with appropriate boundary conditions for the Laplace operator, δ, satisfy a weaker PSA. This weaker PSA is enough to ensure the existence of an inertial manifold for a specific class of scalar-valued reaction diffusion equations on each domain ωn under suitable conditions.

Let a1… ad be a basis of the Lie algebra g of a connected Lie group G and let M be a Lie subgroup of,G. If dx is a non-zero positive quasi-invariant regular Borel measure on the homogeneous space X = G/M and S: X × G → C is a continuous cocycle, then under a rather weak condition on dx and S there exists in a natural way a (weakly*) continuous representation U of G in Lp (X;dx) for all p ε [1,].

Let Ai be the infinitesimal generator with respect to U and the direction ai, for all i ∈ { 1… d}. We consider n–th order strongly elliptic operators H = ΣcαAα with complex coefficients cα. We show that the semigroup S generated by the closure of H has a reduced heat kernel K and we derive upper bounds for k and all its derivatives.

Let u be a solution of the heat equation which can be written as the difference of two non-negative solutions, and let v be a non-negative solution. A study is made of the behaviour of u(x, t)/v(x, t) as t → 0+. The methods are based on the Gauss-Weierstrass integral representation of solutions on Rn × ]0, a[ and results on the relative differentiation of measures, which are employed in a novel way to obtain several domination, non-negativity, uniqueness and representation theorems.

The aim of this article is to review the progress made in the last few years in the representation theory of solutions of parabolic systems in the sense of Petrowskii.

Consider positive solutions of the one dimensional heat equation. The space variable x lies in (–a, a): the time variable t in (0,∞). When the solution u satisfies (i) u (±a, t) = 0, and (ii) u(·, 0) is logconcave, we give a new proof based on the Maximum Principle, that, for any fixed t > 0, u(·, t) remains logconcave. The same proof techniques are used to establish several new results related to this, including results concerning joint concavity in (x, t) similar to those considered in Kennington [15].

We prove that if u(x, t) is a solution of the one dimensional heat equation and if A u(x, t) is its Appell transform, then u(x, t) has the semi-group (Huygens) property in a domain D if and only if A u(x, t) has the semi-group property in a dual region. We apply this result to simplify and extend some results of Rosenbloom and Widder.

A class of functional differential equations in some Hilbert space are studied. The results are applicable to many quasi-linear parabolic paratial differential equations with (possibly) countably many discrete delays and finitely many distributed delays in the highest order spatial derivatives. For the linear case, an evolution operator on the underline space H is introduced, via which a variation of constant formula for the solution of the equation in the underline space H is derived. Some spectral properties of the generator of the solution semigroup defined on some appropriate space are discussed as well.

Let G be a connected Lie group with Lie algebra g and a1, …, ad an algebraic basis of g. Further let Ai denote the generators of left translations, acting on the Lp-spaces Lp(G; dg) formed with left Haar measure dg, in the directions ai. We consider second-order operators in divergence form corresponding to a quadratic form with complex coefficients, bounded Hölder continuous principal coefficients cij and lower order coefficients ci, c′ii, c0 ∈ L∞ such that the matrix C= (cij) of principal coefficients satisfies the subellipticity condition uniformly over G.

We discuss the hierarchy relating smoothness properties of the coefficients of H with smoothness of the kernel and smoothness of the domain of powers of H on the Lρ-spaces. Moreover, we present Gaussian type bounds for the kernel and its derivatives.

Similar theorems are proved for strongly elliptic operators in non-divergence form for which the principal coefficients are at least once differentiable.

We derive upper Gaussian bounds for the heat kernel on complete, noncompact locally symmetric spaces M=Γ∖X with nonpositive curvature. Our bounds contain the Poincaré series of the discrete group Γ and therefore we also provide upper bounds for this series.

This paper deals the global existence and blow-up properties of the following non-Newton polytropic filtration system with nonlocal source, Under appropriate hypotheses, we prove that the solution either exists globally or blows up in finite time depending on the initial data and the relations between αβ and mn(p−1)(q−1). In the special case, α=n(q−1), β=m(p−1), we also give a criteria for the solution to exist globally or blow up in finite time, which depends on a,b and ζ(x),ϑ(x) as defined in our main results.