Well-posedness results for a class of semilinear time-fractional diffusion equations
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Zeitschrift f¨ ur angewandte Mathematik und Physik ZAMP
Well-posedness results for a class of semilinear time-fractional diffusion equations Bruno de Andrade, Vo Van Au, Donal O’Regan and Nguyen Huy Tuan
Abstract. In this paper, we discuss an initial value problem for the semilinear time-fractional diffusion equation. The local well-posedness (existence and regularity) is presented when the source term satisfies a global Lipschitz condition. The unique continuation of solution and finite time blowup result are presented when the reaction terms are logarithmic functions (local Lipschitz types). Mathematics Subject Classification. 26A33, 33E12, 35B40, 35K70, 44A20. Keywords. Well-posedness, Blowup, Fractional calculus, Nonlinear problem.
1. Introduction Let Ω ⊂ RN , (N ≥ 1) be a bounded open set with boundary Ωc . The main aim of the paper is to study the properties of the solutions of a class of time-fractional diffusion equations involving the so-called Riemann–Liouville (R–L) time-fractional derivative. More precisely, we consider the following initial value problem: ⎧ ∂ ∂ 1−α ⎪ ⎪ Au(x, t) + F (u), x ∈ Ω, 0 < t ≤ T, ⎨ u(x, t) = − ∂t ∂t (P) u(x, t) = 0, x ∈ Ωc , 0 < t < T, ⎪ ⎪ ⎩ x ∈ Ω, u(x, 0) = u0 (x), where T > 0, α ∈ (0, 1) is real number and 1 − α of the function u formally given by
∂ 1−α u denotes the R–L time-fractional derivative of order ∂t
∂ 1−α d f (t) := t (J α f ) (t), ∂t d
t > 0,
(1.1)
where the Riemann–Liouville fractional integral operator J α : L2 (0, T ) → L2 (0, T ) is defined by the formula (see, e.g., [1]) ⎧ t ⎪ ⎪ ⎨ 1 τ α−1 f (t − τ )dτ, 0 < α < 1, (1.2) (J α f )(t) := Γ(α) ⎪ 0 ⎪ ⎩ f (t), α = 0, and Γ(·) is the Gamma function. The operator A is a linear, positive definite, self-adjoint operator with compact inverse in L2 (Ω), u = u(x, t) is the state of the unknown function and u0 (x) is a given function. The function F is a nonlinear source term which appears in some physical phenomena [2–4]. 0123456789().: V,-vol
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When α = 1 and A = −Δ problem P describes the nonlinear heat Eq. [2,5–7] ∂ u(x, t) − Δu(x, t) = F (u). (1.3) ∂t If α ∈ (0, 1), Problem P is called an initial value problem for the semilinear time-fractional diffusion equation; we refer the reader to [3,8–10] and the references therein. Many important physical models and practical problems require one to consider the diffusion model with a fractional derivative rather than a classical one, like physical models considering memory effects [2–4,11–13] and some corresponding engineering problems [2,3,14,15] with power-law memory (non-local effects) in time [4,8,16–20]. For nonlinearities of power-type F (u) = |u|p−1 u for p ≥ 1, Bruno de Andrade et al. [3] considered the fractional reaction–diffusion equation to discuss the global well-posedness and asymptotic behavior of solutions; see also [7,21] and the references therein. Studies of logarithmic nonlinearity have a long history in physics as they occur naturally in inflation cosmology, quantum mechanics, and nuclear physics [22] and PDEs with
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