Elliptic equations with diffusion parameterized by the range of nonlocal interactions

Abstract

Abstract. We consider quasilinear elliptic equations where the diffusion at each point depends on all the values of the solution in a neighborhood of this point. The size of this neighborhood is parameterized by some non negative number which represents the range of nonlocal interactions. For fixed values of the parameter, the issue of the existence and local uniqueness of the solution is addressed. In a radial symmetric setting, we give pointwise estimates of the solutions and prove the existence of multiple solutions. Regarding bifurcation theory, we show that many local branches of solutions may exist while, among them, only one is global and has no bifurcation point.

Long-time behavior of a non-autonomous parabolic equation with nonlocal diffusion and sublinear terms

“…For short, we will denote u n (·) = u(·; τ n , u τn ). Thanks to Lemma 20,(2), and (3), we know that there exists τ 1 ( D, t) < t − 2 satisfying that,…”

“…If u is a weak solution to (1), then (2), (3), and (5) imply that u ′ ∈ L 2 (τ, T ; H −1 (Ω)) for any T > τ , and therefore u ∈ C([τ, +∞); L 2 (Ω)). Hence the initial datum in (1) makes sense.…”

“…Lemma 25. Assume that the function a is locally Lipschitz and satisfies (2) (4), and (17), and h ∈ L 2 loc (R; L 2 (Ω)) satisfies (18) for some µ ∈ (0, 2(λ 1 m − α)). Then,…”

“…Namely, we have the following Proposition 19. Assume that the function a is locally Lipschitz and satisfies (2), f ∈ C(R) fulfills (3), (4), and (17)…”

“…In [1,2] the authors analyze the case in which f is still independent of u, and the nonlocal operator is not acting globally in the whole domain but in the part of it contained in a ball centered on each position point. Radial solutions, bifurcation analysis, branch of solutions and their stability are studied.…”