Decay estimates for nonlinear nonlocal diffusion problems in the whole space |
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Authors: | L. I. Ignat D. Pinasco J. D. Rossi A. San Antolin |
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Affiliation: | 1. Institute of Mathematics “Simion Stoilow” of the Romanian Academy, 21 Calea Grivitei Street, 010702, Bucharest, Romania 2. Faculty of Mathematics and Computer Science, University of Bucharest, 14 Academiei Str., 010014, Bucharest, Romania 3. Departamento de Matemáticas y Estadística, Universidad Torcuato di Tella, Mi?ones 2177, C1428ATG, Ciudad Autónoma de Buenos Aires, Argentina 4. Consejo Nacional de Investigaciones Científicas y Técnicas (CONICET), Ciudad Autónoma de Buenos Aires, Buenos Aires, Argentina 5. Departamento de Análisis Matemático, Universidad de Alicante, Ap. Correos 99, 03080, Alicante, Spain 6. Dpto. De Matemáticas, FCEyN, Universidad de Buenos Aires, 1428, Buenos Aires, Argentina
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Abstract: | In this paper, we obtain bounds for the decay rate in the L r (? d )-norm for the solutions of a nonlocal and nonlinear evolution equation, namely, $$u_t left( {x,t} right) = int_{mathbb{R}^d } {Kleft( {x,y} right)left| {uleft( {y,t} right) - uleft( {x,t} right)} right|^{p - 2} left( {uleft( {y,t} right) - uleft( {x,t} right)} right)dy, x in mathbb{R}^d , t > 0.}$$ . We consider a kernel of the form K(x, y) = ψ(y?a(x)) + ψ(x?a(y)), where ψ is a bounded, nonnegative function supported in the unit ball and a is a linear function a(x) = Ax. To obtain the decay rates, we derive lower and upper bounds for the first eigenvalue of a nonlocal diffusion operator of the form $$Tleft( u right) = - int_{mathbb{R}^d } {Kleft( {x,y} right)left| {uleft( y right) - uleft( x right)} right|^{p - 2} left( {uleft( y right) - uleft( x right)} right)dy, 1 leqslant p < infty .}$$ . The upper and lower bounds that we obtain are sharp and provide an explicit expression for the first eigenvalue in the whole space ? d : $$lambda _{1,p} left( {mathbb{R}^d } right) = 2left( {int_{mathbb{R}^d } {psi left( z right)dz} } right)left| {frac{1} {{left| {det A} right|^{1/p} }} - 1} right|^p .$$ Moreover, we deal with the p = ∞ eigenvalue problem, studying the limit of λ 1,p 1/p as p→∞. |
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