Abstract
In this paper, we consider the nonlinear ultraparabolic equation
{partial derivative(t)u + partial derivative(s)u - Delta u = vertical bar u vertical bar(p), t > 0, s > 0, x is an element of D-c, u(t, s, x) = f(x), t > 0, s > 0, x is an element of partial derivative D, u(t, 0, x) = u(1)(t, x), t > 0, x is an element of D-c, u(0, s, x) = u(2)(s, x), s> 0, x is an element of D-c,
where D = (B(0, 1) over bar is the closed unit ball in R-N, N >= 2, D-c is its complement, p > 1, u(i) >= 0, i = 1, 2, and integral(partial derivative D) f(x)dS(x) > 0. We derive the critical exponent for the considered problem in the sense of Fujita. We discuss separately the cases N = 2 and N >= 3. To the best of our knowledge, this is the first work dealing with the blow-up of solutions to multi -time equations in an exterior domain. (C) 2019 Elsevier Inc. All rights reserved.