Nonlinear two-term time fractional diffusion-wave problem

Abstract We find the upper viscosity solutions to a nonlinear two-term time fractional diffusion-wave equation with time operator in the Caputo–Dzherbashyan sense and a nonlinear Lipschitz force term F ∈ L l o c ∞ ( [ 0 , T ) × R ) , T > 0 , x ∈ R , (1) b 1 D ∗ β 1 u ( x , t ) + b 2 D ∗ β 2 u ( x , t ) = ∂ 2 ∂ x 2 u ( x , t ) + F ( t , u ( x , t ) ) , t ≥ 0 , b 1 + b 2 = 1 , β 1 β 2 ∈ ( 0 , 2 ) , subject to the Cauchy conditions (2) u ( x , 0 ) = f ( x ) , u t ( x , 0 ) = g ( x ) , where f , g ∈ L p ( R ) , 1 ≤ p ≤ ∞ . In order to prove the existence and the uniqueness of the solution to this problem we consider first the corresponding linear one. Then, we linearize problem (1) using the first approximation to the nonlinear term F . As a framework we take L p ( R ) -spaces, 1 ≤ p ≤ ∞ .