Heat kernel expansions on the integers and the Toda lattice hierarchy

We consider the heat equation u _t = Lu where L is a second-order difference operator in a discrete variable n. The fundamental solution has an expansion in terms of the Bessel functions of imaginary argument. The coefficients α _k (n, m) in this expansion are analogs of Hadamard’s coefficients for the (continuous) Schr?dinger operator. We derive an explicit formula for α _k in terms of the wave and the adjoint wave functions of the Toda lattice hierarchy. As a first application of this result, we prove that the values of these coefficients on the diagonals n = m and n = m + 1 define a hierarchy of differential-difference equations which is equivalent to the Toda lattice hierarchy. Using this fact and the correspondence between commutative rings of difference operators and algebraic curves we show that the fundamental solution can be summed up, giving a finite formula involving only two Bessel functions with polynomial coefficients in the time variable t, if and only if the operator L belongs to the family of bispectral operators constructed in [18].      

Existence results for a class of porous medium type equations with a quadratic gradient term

In this paper we study the Dirichlet problem in Q _T = Ω × (0, T) for degenerate equations of porous medium-type with a lower order term:

The blow-up property for a system of heat equations with nonlinear boundary conditions

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Fast diffusion flow on manifolds of nonpositive curvature

We consider the fast diffusion equation (FDE) u _t = Δu ^m (0 < m < 1) on a nonparabolic Riemannian manifold M. Existence of weak solutions holds. Then we show that the validity of Euclidean–type Sobolev inequalities implies that certain L ^p −L ^q smoothing effects of the type ∥u(t)∥ _q ≤ Ct ^−α ∥u ₀∥^γ _p , the case q = ∞ being included. The converse holds if m is sufficiently close to one. We then consider the case in which the manifold has the addition gap property min σ(−Δ) > 0. In that case solutions vanish in finite time, and we estimate from below and from above the extinction time.      

带非局部源的退化半线性抛物方程的解的爆破性质

This paper deals with the blow-up properties of the positive solutions to the nonlocal degenerate semilinear parabolic equation u _t − (x ^a u _x ) _x =∫ ₀ ^a f(u)dx in (0,a) × (0,T) under homogeneous Dirichlet conditions. The local existence and uniqueness of classical solution are established. Under appropriate hypotheses, the global existence and blow-up in finite time of positve solutions are obtained. It is also proved that the blow-up set is almost the whole domain. This differs from the local case. Furthermore, the blow-up rate is precisely determined for the special case: f(u)=u ^p , p>1.     

Global existence and L ∞ estimates of solutions for a quasilinear parabolic system

In this paper, we study the global existence, L^∞ estimates and decay estimates of solutions for the quasilinear parabolic system u_t = div (|∇ u|^m ∇ u) + f(u, v), v_t = div (|∇ v|^m ∇ v) + g(u,v) with zero Dirichlet boundary condition in a bounded domain Ω ⊂ R^N. In particular, we find a critical value for the existence and nonexistence of global solutions to the equation u_t = div (|∇ u|^m ∇ u) + λ |u|^{α - 1} u.     

Three-term asymptotics for the boundary layers of semilinear elliptic eigenvalue problems

We consider the nonlinear eigenvalue problem −Δu=λ f(u) in Ω u=0 on ∂Ω, where Ω is a ball or an annulus in R^N (N ≥ 2) and λ > 0 is a parameter. It is known that if λ >> 1, then the corresponding positive solution u_λ develops boundary layers under some conditions on f. We establish the asymptotic formulas for the slope of the boundary layers of u_λ with the exact second term and the ‘optimal’ estimate of the third term.     

Localization of blow-up points for a parabolic system with a nonlinear boundary condition

The authors localize the blow-up points of positive solutions of the systemu _t =Δu,v _t =Δv with conditions at the boundary of a bounded smooth domain Θ under some restrictions off andg and the initial data (Δu ₀, Δν₀>c>0). If Θ is a ball, the hypothesis on the initial data can be removed. Supported by Universidad de Buenos Aires under grant EX071 and CONICET.     

Life span of solutions of a weakly coupled parabolic system

We consider the Cauchy problem for the weakly coupled parabolic system ∂ _t w _λ−Δ w _λ = F(w _λ) in R ^N , where λ > 0, w _λ = (u _λ, v _λ), F(w _λ) = (v _λ ^p , u _λ ^q ) for some p, q ≥ 1, pq > 1, and , for some nonnegative functions φ₁, φ₂ C ₀(R ^N ). If (p, q) is sub-critical or either φ₁ or φ₂ has slow decay at ∞, w _λ blows up for all λ > 0. Under these conditions, we study the blowup of w _λ for λ small.      

Asymptotics of the porous media equation via Sobolev inequalities

Let M be a compact Riemannian manifold without boundary. Consider the porous media equation , u(0)=u₀∈L^q, ? being the Laplace-Beltrami operator. Then, if q?2∨(m-1), the associated evolution is L^q-L^∞ regularizing at any time t>0 and the bound ‖u(t)‖_∞?C(u₀)/t^β holds for t<1 for suitable explicit C(u₀),γ. For large t it is shown that, for general initial data, u(t) approaches its time-independent mean with quantitative bounds on the rate of convergence. Similar bounds are valid when the manifold is not compact, but u(t) approaches u≡0 with different asymptotics. The case of manifolds with boundary and homogeneous Dirichlet, or Neumann, boundary conditions, is treated as well. The proof stems from a new connection between logarithmic Sobolev inequalities and the contractivity properties of the nonlinear evolutions considered, and is therefore applicable to a more abstract setting.     

Asymptotic behavior of viscosity solutions for a degenerate parabolic equation associated with the infinity-Laplacian

The asymptotic behavior of viscosity solutions to the Cauchy–Dirichlet problem for the degenerate parabolic equation u _t  = Δ_∞ u in Ω × (0,∞), where Δ_∞ stands for the so-called infinity-Laplacian, is studied in three cases: (i) and the initial data has a compact support; (ii) Ω is bounded and the boundary condition is zero; (iii) Ω is bounded and the boundary condition is non-zero. Our method of proof is based on the comparison principle and barrier function arguments. Explicit representations of separable type and self-similar type of solutions are also established. Moreover, in case (iii), we propose another type of barrier function deeply related to a solution of . Goro Akagi was supported by the Shibaura Institute of Technology grant for Project Research (no. 2006-211459, 2007-211455), and the grant-in-aid for young scientists (B) (no. 19740073), Ministry of Education, Culture, Sports, Science and Technology. Petri Juutinen was supported by the Academy of Finland project 108374. Ryuji Kajikiya was supported by the grant-in-aid for scientific research (C) (no. 16540179), Ministry of Education, Culture, Sports, Science and Technology.     

The integrable nonlinear degenerate diffusion equationu t=[f(u)u x −1 x and its relatives

The nonlinear diffusion equationu _t=[f(u)g(u _x )] arises in recent models of turbulent transport and of stress dissipation in rock blasting. A Lie point symmetry analysis produces many similarity reductions of exponential and power-law forms, and reveals that for all choices off the equation is always integrable wheng(u _x )=1/u _x . We identify the functionsf(u) which guarantee equivalence to the linear heat equation. For all other choices off, the linear canonical form leads to a self-adjoint differential equation by separation of variablesx andt. We construct a number of explicit solutions with simple boundary conditions, which illustrate behavior in the vicinity of the degenerate region withu _x =. If zero flux and constant concentration are maintained on free boundaries, then steep concentration gradients may evolve from smooth initial conditions. For other boundary conditions, unlike the examples of strong degeneracy, smoothing will occur at initial step discontinuities.     

A Concentration Theorem for the Heat Equation

 We compare the solution of to the solution of the same equation where f is replaced by a “concentrated” source . As a result we derive some estimates on the solution in spatial norm, locally uniformly in t, with respect to the norm of for any integer . In the case we obtain a critical inequality relating the norm of to an exponential norm of u. (Received 1 September 2000; in revised form 17 January 2001)     

Solution of two-dimensional boundary value problems corresponding to initial-boundary value problems of diffusion on a right cylinder

We consider boundary value problems for the differential equations Δ₂ u + B u = 0 with operator coefficients B corresponding to initial-boundary value problems for the diffusion equation Δ₃ u − pu = ∂ _t u (p > 0) on a right cylinder with inhomogeneous boundary conditions on the lateral surface of the cylinder with zero boundary conditions on the bases of the cylinder and with zero initial condition. For their solution, we derive specific boundary integral equations in which the space integration is performed only over the lateral surface of the cylinder and the kernels are expressed via the fundamental solution of the two-dimensional heat equation and the Green function of corresponding one-dimensional initial-boundary value problems of diffusion. We prove uniqueness theorems and obtain sufficient existence conditions for such solutions in the class of functions with continuous L ₂-norm.     

The Dirichlet Problem for the Porous Medium Equation in Bounded Domains. Asymptotic Behavior

The paper contains a complete study of the large-time behavior of solutions of the Porous Medium Equation, u_t = u^m with m > 1, posed in a bounded domain of the n-dimensional space with homogeneous boundary conditions. Asymptotic profiles are obtained and full proofs of the convergence results are given. Solutions with changing sign and problems posed in tubes are also considered.     

The initial boundary value problem for the nonlinear parabolic equation in unbounded domain

In this paper we consider the mixed problem for the equation u _tt  + A ₁ u + A ₂(u _t ) + g(u _t ) = f(x, t) in unbounded domain, where A ₁ is a linear elliptic operator of the fourth order and A ₂ is a nonlinear elliptic operator of the second order. Under natural assumptions on the equation coefficients and f we proof existence of a solution. This result contains, as a special case, some of known before theorems of existence. Essentially, in difference up to previous results we prove theorems of existence without the additional assumption on behavior of solution at infinity.      

Decay rate of the range component of solutions to some semilinear evolution equations

We examine the rate of decay to 0, as t → +∞., of the projection on the range of A of the solutions of an equation of the form u′ + Au + |u| ^p−1 u = 0 or u′′ + u′ + Au + |u| ^p−1 u = 0 in a bounded domain of ^N , where A = −Δ with Neumann boundary conditions or A = −Δ − λ₁ I with Dirichlet boundary conditions. In general this decay is much faster than the decay of the projection on the kernel; it is often exponential, but apparently not always.     

Periodic parabolic problems with concave and convex nonlinearities

Let be a smooth bounded domain, let a, b be two functions that are possibly discontinuous and unbounded with a ≥ 0 in and b > 0 in a set of positive measure and let 0 < p < 1 < q. We prove that there exists some 0 < Λ < ∞ such that the nonlinear Dirichlet periodic parabolic problem in has a positive solution for all 0 < λ < Λ and that there is no positive solution if λ > Λ. In some cases we also show the existence of a minimal solution for all 0 < λ < Λ and that the solution u _λ can be chosen such that λ → u _λ is differentiable and increasing. We also give some upper and lower estimates for such a Λ. All results remain true for the analogous elliptic problems. Partially supported by CONICET, Secyt-UNC, ANPCYT and Agencia Cordoba Ciencia     

A bifurcation problem governed by the boundary condition I

We deal with positive solutions of Δu = a(x)u ^p in a bounded smooth domain subject to the boundary condition ∂u/∂v = λu, λ a parameter, p > 1. We prove that this problem has a unique positive solution if and only if 0 < λ < σ₁ where, roughly speaking, σ₁ is finite if and only if |∂Ω ∩ {a = 0}| > 0 and coincides with the first eigenvalue of an associated eigenvalue problem. Moreover, we find the limit profile of the solution as λ → σ₁. Supported by DGES and FEDER under grant BFM2001-3894 (J. García-Melián and J. Sabina) and ANPCyT PICT No. 03-05009 (J. D. Rossi). J.D. Rossi is a member of CONICET.     

A note on the behavior of blow-up solutions for one-phase Stefan problems

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