Large time behaviour of solutions of a fast diffusion equation with source

1994 ◽  
Vol 23 (12) ◽  
pp. 1559-1568 ◽  
Author(s):  
Jong-Shenq Guo
Keyword(s):  
Diffusion Equation ◽  
Large Time ◽  
Fast Diffusion ◽  
Time Behaviour ◽  
Fast Diffusion Equation ◽  
Large Time Behaviour

scholarly journals Large-time behaviour of the higher-dimensional logarithmic diffusion equation

2013 ◽  
Vol 143 (4) ◽  
pp. 817-830 ◽  
Author(s):  
Kin Ming Hui ◽  
Sunghoon Kim
Keyword(s):  
Diffusion Equation ◽  
Global Solution ◽  
Compact Subset ◽  
Symmetric Solution ◽  
Large Time ◽  
Time Behaviour ◽  
Radially Symmetric Solution ◽  
Large Time Behaviour ◽  
Higher Dimensional

Let n ≥ 3 and let ψλ0 be the radially symmetric solution of Δ log ψ + 2βψ + βx · ∇ψ = 0 in ℝn, ψ(0) = λ(0), for some constants λ0 > 0, β > 0. Suppose u0 ≥ 0 satisfies u0 − ψλ0 ∈ L1 (ℝn) and u0 (x) ≈ (2(n − 2)/β)(log∣x∣/∣x∣2) as ∣x∣ → ∞. We prove that the rescaled solution ũ(x,t) = e2βtu(eβtx, t) of the maximal global solution u of the equation ut = Δ log u in ℝn × (0, ∞), u(x, 0) = u0 (x) in ℝn, converges uniformly on every compact subset of ℝn and in L1 (ℝn) to ψλ0 as t → ∞. Moreover, ∥ũ(·, t) − ψλ0∥L1(ℝn) ≤ e−(n−2)βt∥u0 − ψλ0∥L1(ℝn) for all t ≥ 0.

scholarly journals The asymptotic behaviour of Ramanujan's integral and its application to two-dimensional diffusion-like equations

2000 ◽  
Vol 11 (1) ◽  
pp. 13-28 ◽  
Author(s):  
STEFAN G. LLEWELLYN SMITH
Keyword(s):  
Diffusion Equation ◽  
Large Time ◽  
Domain Boundary ◽  
Radial Symmetry ◽  
Two Dimensional ◽  
Time Behaviour ◽  
Simple Method ◽  
Linear Diffusion ◽  
Large Time Behaviour ◽  
Logarithmic Decay

The large-time behaviour of a large class of solutions to the two-dimensional linear diffusion equation in situations with radial symmetry is governed by the function known as Ramanujan's integral. This is also true when the diffusion coefficient is complex, which corresponds to Schrödinger's equation. We examine the asymptotic expansion of Ramanujan's integral for large values of its argument over the whole complex plane by considering the analytic continuation of Ramanujan's integral to the left half-plane. The resulting expansions are compared to accurate numerical computations of the integral. The large-time behaviour derived from Ramanujan's integral of the solution to the diffusion equation outside a cylinder is not valid far from the domain boundary. A simple method based on matched asymptotic expansions is outlined to calculate the solution at large times and distances: the resulting form of the solution combines the inverse logarithmic decay in time typical of Ramanujan's integral with spatial dependence on the usual similarity variable for the diffusion equation.

scholarly journals Large time behavior for the fast diffusion equation with critical absorption

2016 ◽  
Vol 260 (11) ◽  
pp. 8000-8024 ◽  
Author(s):  
Said Benachour ◽  
Razvan Gabriel Iagar ◽  
Philippe Laurençot
Keyword(s):  
Diffusion Equation ◽  
Large Time ◽  
Large Time Behavior ◽  
Fast Diffusion ◽  
Time Behavior ◽  
Fast Diffusion Equation
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