On positive solutions of the Cauchy problem for doubly nonlocal equations

We study the Cauchy problem for the doubly nonlocal equation ut=Du+parallel to K(u)q parallel to(p-1/q)(L1(Rn))u(1+r), (0.1) where Du=J*u-u,J similar to |x|(-n-alpha) and K similar to |x|(-m)as|x| -> infinity and for some m is an element of R and 0 < alpha <= +infinity, here alpha = +infinity means that J is compactly supported. The problem not only describes nonlocal diffusions and nonlocal interactions, but also it is a nonlocal counterpart of the nonlocal heat equation in [10] where and m > 0. As for the local existence of positive solutions, we find that problem (0.1) is quite different from the nonlocal heat equation. That is, problem (0.1) admits positive solutions if and only if m > n - q(n + alpha), however the nonlocal heat equation has positive solutions for any m is an element of R. For m >= 0 or n(1 - q) < m < 0 with alpha = +infinity, we determine the critical Fujita curve of problem (0.1) as F(p, q, n, m, r, alpha) := (p - 1)[nq - (n - m)+] - q(beta - nr) = 0 with beta = min{alpha, 2}. That is, if F <= 0, every positive solution of (0.1) blows up in finite time, whereas if F > 0, there exist both nonglobal solutions and global solutions. In the supercritical case F > 0, we further discuss the second critical exponent which describes the critical decay rate of initial data to distinguish both solutions. Particularly, we find a new phenomenon, namely, if n(1-q)< m <= n-q beta/r with alpha = +infinity, the second critical exponent is independent of p and r.