An inhomogeneous nonlocal diffusion problem with unbounded steps
Tóm tắt
We consider the following nonlocal equation
$$\int J\left(\frac{x-y}{g(y)}
\right) \frac{u(y)}{g(y)} dy -u(x)=0\qquad x\in \mathbb{R},$$
where J is an even, compactly supported, Hölder continuous kernel with unit integral and g is a continuous positive function. Our main concern will be with unbounded functions g, contrary to previous works. More precisely, we study the influence of the growth of g at infinity on the integrability of positive solutions of this equation, therefore determining the asymptotic behavior as
$${t\to +\infty}$$
of the solutions to the associated evolution problem in terms of the growth of g.
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