Global existence and blow-up for a mixed pseudo-parabolic p-Laplacian type equation with logarithmic nonlinearity

In this paper, we study an initial -boundary value problem for the following mixed pseudo -parabolic p -Laplacian type equation with logarithmic nonlinearity in a bounded domain Omega subset of R-n. : u(t) - Delta u(t) - div (vertical bar del u vertical bar(p-2) del u) vertical bar u vertical bar(q-2) u log vertical bar u vertical bar, 1 < p <= q { < infinity < np/n--p, if n <= p ; if n <= p which is referred to as Showalter equation [1]. We focus on the initial conditions, which ensure the solutions of the model exist global, blow up in finite time and blow up at infinite time. Explicitly, we classify the ranges of p, q into two cases: 1 < p <= q {<= 2, < if n <= p, <= 2, < if 2n/n+2 <p < n < np/n-p , if p <= 2n/n+2, (0.1) and 1 < p <= q < 2, < { infinity/ np/n-p, if n <= p, if 2n+/n+2 < p < n. (0.2) Let J(u(0)) be the initial energy (see (2.1) for the definition of J), I(u(0)) be the vale of Nehari functional at uo (see (2.2) for the definition of I), d > 0 be the mountain -pass level given in (2.4), and M E [0, d] be the constant given in (2.6), we get (i) If J(u(0)) <= d and /(u(0)) >= 0, then the solution exists globally; (ii) If J(u(0)) <= d, I(u(0)) < 0 and p, q satisfies (0.1), then the solution blows up at infinite time. Moreover, if in addition J(u(0) (<= M, if < < d < M, if M = d, the growth rate of the solution is got; (iii) If J(u(0) <= M, I(u(0)) < 0p, q satisfies (0.2), then the solution blows up in finite time. (C)19 Elsevier Inc. All rights reserved.