Temporal monotonicity of the solutions of some semilinear parabolic equations with fractional diffusion

Abstract

Suppose that the functions g , φ and ψ are nonnegative and satisfy suitable regularity conditions. Then, we prove in this work that the parabolic semilinear problem ∂ t u ( t , x ) = Δ α u ( t , x ) − g ( x ) f ( u ( t , x ) ) + φ ( x ) , t > 0 , x ∈ R n , u ( 0 , x ) = ψ ( x ) , x ∈ R n , has a unique positive and time-monotone solution. Here, Δ α is the fractional Laplacian with α ∈ ( 0 , 2 ] , and the source term f is a convex function with f ( 0 ) = 0 . Moreover, using the temporal monotonicity, we show that the elliptic equation Δ α v ( x ) = g ( x ) f ( v ( x ) ) − φ ( x ) , x ∈ R d , with boundary condition lim ‖ x ‖ → ∞ v ( x ) = 0 , has a positive solution. We provide also sufficient conditions for the integrability of both solutions.