On a nonlinear parabolic equation involving Besselʹs operator associated with a mixed inhomogeneous condition

In this paper we consider the following nonlinear parabolic equation(*) u t - a ( t ) u rr + γ r u r + F ( r , u ) = f ( r , t ) , 0 < r < 1 , 0 < t < T , lim r → 0 + r γ / 2 u r ( r , t ) < + ∞ , u r ( 1 , t ) + h ( t ) ( u ( 1 , t ) - u ˜ 0 ) = 0 , u ( r , 0 ) = u 0 ( r ) , where γ > 0 , u ˜ 0 are given constants, a ( t ) , h ( t ) , F ( r , u ) , f ( r , t ) are given functions. In Section 3, we use the Galerkin and compactness method in appropriate Sobolev spaces with weight to prove the existence of a unique weak solution of the problem (*) on ( 0 , T ) , for every T > 0 . In Section 4, we prove that if the initial condition is bounded, then so is the solution. In Section 5, we study asymptotic behavior of the solution as t → + ∞ . In Section 6 we give numerical results.