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.

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.

This work is concerned with the problem posed in the domain which is not necessary rectangular, and with Our goal is to find some conditions on the coefficient c and the functions ( φi) i=1,2 such that the solution of this problem belongs to the Sobolev space

For a large class of operator inclusions, including those generated by maps of pseudomonotone type, we obtain a general theorem on existence of solutions. We apply this result to some particular examples. This theorem is proved using the method of difference approximations.

We investigate the conditions on a hedger, who overestimates the (time- and level-dependent) volatility, to superreplicate a convex claim on several underlying assets. It is shown that the classic Black-Scholes model is the only model, within a large class, for which overestimation of the volatility yields the desired superreplication property. This is in contrast to the one-dimensional case, in which it is known that overestimation of the volatility with any time- and level-dependent model guarantees superreplication of convex claims.

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.

In this article, we analyse the error induced by the Euler scheme combined with a symmetry procedure near the boundary for the simulation of diffusion processes with an oblique reflection on a smooth boundary. This procedure is easy to implement and, in addition, accurate: indeed, we prove that it yields a weak rate of convergence of order 1 with respect to the time-discretization step.

Using a representation in terms of a two-type branching particle system, we prove that positive solutions of the system remain bounded for suitable bounded initial conditions, provided A and B generate processes with independent increments and one of the processes is transient with a uniform power decay of its semigroup. For the case of symmetric stable processes on R1,this answers a question raised in [4].

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.

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.

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 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 .

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.

In Section 1 of this note we will construct an example of a subset of R × Rn such that the parabolic capacity with respect to the heat equation is zero although its orthogonal projection onto {0} × Rn is the whole space. Such examples were already given by R. Kaufman and J.-M. Wu in [5] and [6]. However, our probabilistic approach seems to be more transparent since it does not depend on explicit formulas for Green functions.

In this paper, we consider an extended model of a coupled nonlinear reaction-diffusion equations with Neumann-Neumann boundary conditions. We obtain upper linear growth bound for one of the components. We also find the corresponding bound for the case of Dirichlet-Dirichlet boundary conditions.

The Cauchy problem in the form of (1.11) with linear and constant coefficients is discussed. The solution (1.10) can be given in explicit form when the stochastic process is a multidimensional autoregression (AR) type, or Ornstein–Uhlenbeck process. Functionals of (1.10) form were studied by Kac in the Brownian motion case. The solutions are obtained with the help of the Radon–Nikodym transformation, proposed by Novikov [12].