On the nonlinear pseudoparabolic equation with the mixed inhomogeneous condition

Abstract

We study the following initial-boundary value problem: ⎧ ⎨ ⎩ ut – (μ + α ∂ ∂t )( ∂2u ∂x2 + 1 x ∂u ∂x ) + f(u) = f1(x,t), 1 < x < R,t > 0, ux(1,t) = h1u(1,t) + g1(t), u(R,t) = gR(t), u(x, 0) = u˜ 0(x), () where μ > 0, α > 0, h1 ≥ 0, R > 1 are given constants and f, f1, g1, gR, u˜ 0 are given functions. First, we use the Galerkin and compactness method to prove the existence of a unique weak solution u(t) of Problem (1) on (0, T), for every T > 0. Next, we study the asymptotic behavior of the solution u(t) as t → +∞. Finally, we prove the existence and uniqueness of a weak solution of Problem (1)1,2 associated with a ‘(N + 1)-points condition in time’ case, u(x, 0) = N i=1 ηiu(x, Ti), () where (Ti,ηi), i = 1, ... ,N, are given constants satisfying 0 < T1 < T2 < ··· < TN–1 < TN ≡ T, N i=1 |ηi

≤ 1. MSC: 34B60; 35K55; 35Q72; 80Axx Keywords: nonlinear pseudoparabolic equation; Faedo-Galerkin approximation; asymptotic behavior; ‘(N + 1)-points condition in time