Well-posedness, global existence and large time behavior for Hardy-Henon parabolic equations

In this paper we study the nonlinear parabolic equation partial derivative(t)u = Delta u+a vertical bar x vertical bar(-gamma)vertical bar u vertical bar(alpha)u, t > 0, x is an element of R-N \ {0}, N >= 1, a is an element of R, alpha > 0, 0 < gamma < min(2,N) and with initial value u(0) = phi. We establish local well-posedness in L-q(R-N) and in C-0(R-N). In particular, the value q= N alpha/(2-gamma) plays a critical role. For alpha > (2-theta)/N, we show the existence of global self -similar solutions with initial values phi(x) = omega(x)vertical bar x vertical bar(-(2-gamma)/alpha), where omega is an element of L-infinity(R-N) is homogeneous of degree 0 and parallel to omega parallel to(infinity) is sufficiently small. We then prove that if phi(x) similar to omega(x)vertical bar x vertical bar(-(2-gamma)/alpha) for vertical bar x vertical bar large, then the solution is global and is asymptotic in the L-infinity-norm to a self similar solution of the nonlinear equation. While if phi(x) similar to omega(x)vertical bar x vertical bar(-sigma) for vertical bar x vertical bar large with (2-gamma)/alpha < sigma < N, then the solution is global but is asymptotic in the L-infinity-norm to et Delta(omega(omega)vertical bar x vertical bar(-sigma)). The equation with more general potential, partial derivative(t)u = Delta u+V vertical bar x vertical bar(-gamma)vertical bar u vertical bar(alpha)u, V phi(x) similar to omega(x)vertical bar x vertical bar(-(2-gamma)/alpha-gamma) is an element of L-infinity(R-N), is also studied. In particular, for initial data phi(x) similar to omega(x)vertical bar x vertical bar(-(2-gamma)/alpha),vertical bar x vertical bar large, we show that the large time behavior is linear if V is compactly supported near the origin, while it is nonlinear if V is compactly supported near infinity. (C) 2016 Elsevier Ltd. All rights reserved.