Initial value problem

Quantum mechanics

Ecology

Mathematical physics

Biology

Boundary (topology)

Laplace operator

Weak solution

Boundary value problem

Computer science

Bounded function

Type (biology)

Physics

Nonlinear system

Mathematics

Domain (mathematical analysis)

Mathematical analysis

Combinatorics

Logarithm

Programming language

Partial differential equation

p-Laplacian

Parabolic partial differential equation

Constant (computer programming)

Journal of Mathematical Analysis and Applications

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 Ω⊂Rn:ut−Δut−div(|∇u|p−2∇u)=|u|q−2ulog⁡|u|,1<p≤q{<∞,if n≤p;<npn−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:(0.1)1<p≤q{≤2,if n≤p,≤2,if 2nn+2<p<n,<npn−p,if p≤2nn+2, and(0.2)1<p≤q and 2<q<{∞, if n≤p,npn−p, if 2nn+2<p<n. Let J(u0) be the initial energy (see (2.1) for the definition of J), I(u0) be the vale of Nehari functional at u0 (see (2.2) for the definition of I), d>0 be the mountain-pass level given in (2.4), and M∈[0,d] be the constant given in (2.6), we get If J(u0)≤d and I(u0)≥0, then the solution exists globally; If J(u0)≤d, I(u0)<0 and p,q satisfies (0.1), then the solution blows up at infinite time. Moreover, if in additionJ(u0){≤M, if M<d,<M, if M=d, the growth rate of the solution is got; If J(u0)≤M, I(u0)<0 and p,q satisfies (0.2), then the solution blows up in finite time.

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