Nonexistence results for a time-fractional biharmonic diffusion equation

• Published in: Boundary value problems Vol. 2024; no. 1; pp. 66 - 17
• Main Authors: Jleli, Mohamed, Samet, Bessem
• Format: Journal Article
• Language: English
• Published: Cham Springer International Publishing 23.05.2024; Hindawi Limited; SpringerOpen
• Subjects: Analysis; Approximations and Expansions; Boundary conditions; Boundary value problems; Caputo fractional derivative; Difference and Functional Equations; Differential equations; Initial conditions; Local and NonLocal Boundary value Problems; Mathematics; Mathematics and Statistics; Nonexistence; Operators (mathematics); Ordinary Differential Equations; Partial Differential Equations; Time-fractional biharmonic diffusion equation; Weak solution; Weighting functions; Weak solution; Time-fractional biharmonic diffusion equation; Nonexistence; 35A01; 26A33; 35R11; Caputo fractional derivative
• ISSN: 1687-2770; 1687-2762; 1687-2770
• DOI: 10.1186/s13661-024-01874-y

We consider weak solutions of the nonlinear time-fractional biharmonic diffusion equation ∂ t α u + ∂ t β u + u x x x x = h ( t , x ) | u | p in ( 0 , ∞ ) × ( 0 , 1 ) subject to the initial conditions u ( 0 , x ) = u 0 ( x ) , u t ( 0 , x ) = u 1 ( x ) and the Navier boundary conditions u ( t , 1 ) = u x x ( t , 1 ) = 0 , where α ∈ ( 0 , 1 ) , β ∈ ( 1 , 2 ) , ∂ t α (resp. ∂ t β ) is the fractional derivative of order α (resp. β ) with respect to the time-variable in the Caputo sense, p > 1 and h is a measurable positive weight function. Using nonlinear capacity estimates specifically adapted to the fourth-order differential operator ∂ 4 ∂ x 4 , the domain, the initial conditions and the boundary condition, a general nonexistence result is established. Next, some special cases of weight functions h are discussed.