On the nonlinear pseudoparabolic equation with the mixed inhomogeneous condition

We study the following initial-boundary value problem: 1 { u t − ( μ + α ∂ ∂ t ) ( ∂ 2 u ∂ x 2 + 1 x ∂ u ∂ x ) + f ( u ) = f 1 ( x , t ) , 1 < x < R , t > 0 , u x ( 1 , t ) = h 1 u ( 1 , t ) + g 1 ( t ) , u ( R , t ) = g R ( t ) , u ( x , 0 ) = u ˜ 0 ( x ) , where μ > 0 , α > 0 , h 1 ≥ 0 , R > 1 are given constants and f , f 1 , g 1 , g R , 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, 2 u ( x , 0 ) = ∑ i = 1 N η i u ( x , T i ) , where ( T i , η i ) , i = 1 , … , N , are given constants satisfying 0 < T 1 < T 2 < ⋯ < T N − 1 < T N ≡ T , ∑ i = 1 N | η i | ≤ 1 .