We study the asymptotic behavior for nonlocal diffusion models of the form u t = J * u − u in the whole R N or in a bounded smooth domain with Dirichlet or Neumann boundary conditions. In R N we obtain that the long time behavior of the solutions is determined by the behavior of the Fourier transform of J near the origin, which is linked to the behavior of J at infinity., the asymptotic behavior is the same as the one for solutions of the evolution given by the α/2 fractional power of the laplacian. In particular when the nonlocal diffusion is given by a compactly supported kernel the asymptotic behavior is the same as the one for the heat equation, which is a local model. Concerning the Dirichlet problem for the nonlocal model we prove that the asymptotic behavior is given by an exponential decay to zero at a rate given by the first eigenvalue of an associated eigenvalue problem with profile an eigenfunction of the first eigenvalue. Finally, we analyze the Neumann problem and find an exponential convergence to the mean value of the initial condition.and variations of it, have been recently widely used to model diffusion processes, see [1], [3], [6], [9], [11], [16], [17], [20], [21] and [22]. As stated in [16], if u(x, t) is thought of as the density of a single population at the point x at time t, and J(x−y) is thought of as the probability distribution of jumping from location y to location x, then (J * u)(x, t) = R N J(y −x)u(y, t) dy is

We study the blow-up behaviour of solutions of a 2mth order semilinear parabolic equationwith a superlinear function q(u) for |u| 1. We prove some estimates on the asymptotic blow-up behaviour. Such estimates apply to general integral evolution equations. We answer the following question: find a continuous function q(u) with a superlinear growth as u → ∞ such that the parabolic equation exhibits regional blow-up in a domain of finite non-zero measure. We show that such a regional blow-up can occur for q(u) = u| ln |u|| 2m . We present a formal asymptotic theory explaining that the stable (generic) blow-up behaviour as t → T − is described by the self-similar solutionof the complex Hamilton-Jacobi equation Introduction: estimates and regional blow-upWe consider the Cauchy problem for the 2mth order semilinear parabolic equation). Higher-order semilinear and quasilinear diffusion operators occur in several applications, including thin film theory, flame and wave propagation, phase transition at critical Lifschitz points and bistable systems (e.g. the Kuramoto-Sivashinskii equation and the extended Fisher-Kolmogorov equation). See references in the book by Peletier & Troy [23].Formation of singularities in higher-order heat equations, which is well understood for the second order reaction-diffusion equations, is a problem of current interest in the general theory of higher-order parabolic equations.

The entry and persistence of phosphomycin in interstitial tissue fluid (ITF) were studied in 9 patients with normal renal function and 8 patients with varying degrees of renal impairment, all of whom received a single i.v. dose of 30 mg/kg. ITF was obtained from skin blisters produced by suction. The antibiotic followed a two-compartment open kinetic model. In patients with normal renal function, phosphomycin is incorporated rapidly into the ITF reaching a level of 60.4 micrograms/ml 60 min after administration. There was no statistically significant difference between the elimination rates from serum and ITF. The serum half-life of the slow disposition phase was 1.75 h in patients with normal renal function. There was a linear correlation between the elimination half-life of phosphomycin in serum and ITF in subjects with differing degrees of renal impairment.

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We prove the uniqueness of positive Wwhere N ≥ 3, 1 < p <