Weighted L(L)-estimate with Muckenhoupt weights for the diffusion-wave equations with time-fractional derivatives

Han, Beom-Seok; Kim, Kyeong Hun; Park, Daehan

doi:10.1016/j.jde.2020.03.005

Cited by 20 publications

(10 citation statements)

References 11 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Proof. The claim is a consequence of [5,Theorem 3.4] with w 2 (t) = 1 and w 1 (x) = |x 1 | θ−d there. We remark that |x…”

Section: Equation With Constant Coefficients On the Half Spacementioning

confidence: 90%

See 1 more Smart Citation

A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains

Han¹,

Kim²,

Park³

2021

DCDS

Self Cite

View full text Add to dashboard Cite

We introduce a weighted Lp-theory (p > 1) for the time-fractional diffusion-wave equation of the typet denotes the Caputo fractional derivative of order α, and Ω is a C 1 domain in R d . We prove existence and uniqueness results in Sobolev spaces with weights which allow derivatives of solutions to blow up near the boundary. The order of derivatives of solutions can be any real number, and in particular it can be fractional or negative.

show abstract

“…Proof. The claim is a consequence of [5,Theorem 3.4] with w 2 (t) = 1 and w 1 (x) = |x 1 | θ−d there. We remark that |x…”

Section: Equation With Constant Coefficients On the Half Spacementioning

confidence: 90%

“…The spaces H γ p,θ (Ω) are independent of the choice of ψ and ζ, and the norms introduced by other choices of ψ and ζ are all equivalent (see [16,18]) for details). In particular, for any η ∈ C ∞ c (R + ) and γ ∈ R, 5) and the inverse also holds if η satisfies (2.2). It also holds that if γ is a non-negative integer then…”

mentioning

confidence: 99%

A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains

Han¹,

Kim²,

Park³

2021

DCDS

Self Cite

View full text Add to dashboard Cite

show abstract

“…For weighted mixed norm estimate, see [12] for the case a ij = δ ij . The result of [12] is based on solution representation by using kernel, and the authors control the mean oscillation of the solution and its derivative to find weighted mixed estimation. Quite recently, weighted mixed norm estimation under the same condition in [5] is proved in [6].…”

Section: Introductionmentioning

confidence: 99%

“…Quite recently, weighted mixed norm estimation under the same condition in [5] is proved in [6]. However, the results of [6,12] only cover the case γ = 0. Also the result of [12] only covers the case a ij = δ ij , and the result of [6] does not cover the range of α ∈ (1, 2).…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Weighted maximal $L_{q}(L_{p})$-regularity theory for time-fractional diffusion-wave equations with variable coefficients

Park¹

2021

Preprint

Self Cite

View full text Add to dashboard Cite

We present a maximal Lq(Lp)-regularity theory with Muckenhoupt weights for the equationHere, ∂ α t is the Caputo fractional derivative of order α ∈ (0, 2) and a ij are functions of (t, x). Precisely, we show thatwhere 1 < p, q < ∞, γ ∈ R, and w 1 and w 2 are Muckenhoupt weights, and this implies that we prove maximal regularity theory. Our approach is based on Fourier multiplier and singular integral theory with Muckenhoupt weights. Also, we use abstract interpolation theory to find interpolation inequality in weighted Sobolev space.

show abstract

Stability and asymptotics for fractional delay diffusion‐wave equations

Yao

Yang

2023

Math Methods in App Sciences

View full text Add to dashboard Cite

The asymptotic stability and long‐time decay rates of solutions to linear Caputo time‐fractional ordinary differential equations (FODEs) with order both and are known to be completely determined by the eigenvalues of the coefficient matrices. Very different from the exponential decay of solutions to classical ordinary differential equations, solutions of FODEs decay only polynomially, leading to the so‐called Mittag–Leffler stability, which was already extended to linear fractional delay differential equations (FDDEs) with order . However, the study on the long‐time decay rates of the solutions for FDDEs with order is scarce. Hence, this paper is devoted to the asymptotical stability and long‐time decay rates for a class of FDDEs with , that is, fractional delay diffusion‐wave equations. We reformulate the fractional delay diffusion‐wave equations to infinite‐dimensional FDDEs by the eigenvalue decomposition method. Through a detailed analysis for the characteristic equations, a novel necessary and sufficient stability condition is established in a coefficient criterion. Moreover, we are able to deal with the awful singularities caused by delay and fractional exponent by introducing a novel integral path, and hence to give an accurate estimation for the fractional resolvent operators. We show that the long‐time decay rates of the solutions for both linear FDDEs and fractional delay diffusion‐wave equations are . Finally, an example of an application model with feedback control is provided to show the effectiveness of our theoretical results.

show abstract

Weighted L(L)-estimate with Muckenhoupt weights for the diffusion-wave equations with time-fractional derivatives

Cited by 20 publications

References 11 publications

A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains

A weighted Sobolev space theory for the diffusion-wave equations with time-fractional derivatives on $ C^{1} $ domains

Weighted maximal $L_{q}(L_{p})$-regularity theory for time-fractional diffusion-wave equations with variable coefficients

Stability and asymptotics for fractional delay diffusion‐wave equations

Contact Info

Product

Resources

About