Optimal existence and uniqueness theory for the fractional heat equation

Bonforte, Matteo; Sire, Yannick; Vázquez, Juan Luís

doi:10.1016/j.na.2016.08.027

Cited by 89 publications

(71 citation statements)

References 45 publications

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“…The analysis of the linear semigroup generated by shows that the first term in the asymptotic behavior may be chosen as

M K_{t}^{α}

where

K_{t}^{α}

is the fundamental solution of problem . See for instance [, Theorem 6.3]. In Subsection 2.2 we present more details about the linear model and its properties.…”

Section: Preliminariesmentioning

confidence: 99%

“…We recall some useful results concerning the associated diffusion problem , that is the Fractional Heat Equation for

0 < α < 2

. We consider the initial value problem

({, \begin{matrix} U_{t} false(t, x false) + {(- Δ)}^{α / 2} U false(t, x false) = 0 for x \in R and t > 0, \\ U false(0, x false) = U_{0} false(x false) for x \in R . \end{matrix})

This problem has been widely studied and many results are known (see for the probabilistic point of view, for a nice motivation of the model and the recent survey for a complete characterization). Some useful properties are proved in [, Section 2].…”

Section: Preliminariesmentioning

confidence: 99%

“…For any

α \in (0, 2)

the profile

F_{α}

C^{\infty} false(double-struckR false)

, positive and decreasing on

(0, \infty)

, and behaves at infinity like

F_{α} false(r false) \sim r^{- (1 + α)}

. Moreover, the solution of Problem behaves as time

t \to \infty

M K_{t}^{α}

, where

0true M = \int_{R} U_{0} (x) d x

is the total mass:

t^{\frac{1}{α} (1 - \frac{1}{p})} {∥ U false(t, \cdot false) - M K_{t}^{α} false(\cdot false) ∥}_{L^{p} false(double-struckR false)} \to 0 as t \to \infty .

See for instance [, Theorem 6.3]. Throughout the paper we will need the following time decay estimates on the fractional derivatives of the kernel.…”

Section: Preliminariesmentioning

confidence: 99%

“…This problem has been widely studied and many results are known (see [4,5,9] for the probabilistic point of view, [41] for a nice motivation of the model and the recent survey [10] for a complete characterization). Some useful properties are proved in [18,Section 2].…”

Section: Reminder On Linear Fractional Diffusionmentioning

confidence: 99%

See 3 more Smart Citations

Asymptotic behavior of solutions to fractional diffusion–convection equations

Ignat

Stan

2018

J. London Math. Soc.

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“…The analysis of the linear semigroup generated by shows that the first term in the asymptotic behavior may be chosen as

M K_{t}^{α}

where

K_{t}^{α}

is the fundamental solution of problem . See for instance [, Theorem 6.3]. In Subsection 2.2 we present more details about the linear model and its properties.…”

Section: Preliminariesmentioning

confidence: 99%

“…We recall some useful results concerning the associated diffusion problem , that is the Fractional Heat Equation for

0 < α < 2

. We consider the initial value problem

({, \begin{matrix} U_{t} false(t, x false) + {(- Δ)}^{α / 2} U false(t, x false) = 0 for x \in R and t > 0, \\ U false(0, x false) = U_{0} false(x false) for x \in R . \end{matrix})

Section: Preliminariesmentioning

confidence: 99%

“…For any

α \in (0, 2)

the profile

F_{α}

C^{\infty} false(double-struckR false)

, positive and decreasing on

(0, \infty)

, and behaves at infinity like

F_{α} false(r false) \sim r^{- (1 + α)}

. Moreover, the solution of Problem behaves as time

t \to \infty

M K_{t}^{α}

, where

0true M = \int_{R} U_{0} (x) d x

is the total mass:

t^{\frac{1}{α} (1 - \frac{1}{p})} {∥ U false(t, \cdot false) - M K_{t}^{α} false(\cdot false) ∥}_{L^{p} false(double-struckR false)} \to 0 as t \to \infty .

See for instance [, Theorem 6.3]. Throughout the paper we will need the following time decay estimates on the fractional derivatives of the kernel.…”

Section: Preliminariesmentioning

confidence: 99%

Section: Reminder On Linear Fractional Diffusionmentioning

confidence: 99%

See 2 more Smart Citations

Asymptotic behavior of solutions to fractional diffusion–convection equations

Ignat

Stan

2018

J. London Math. Soc.

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show abstract

“…We can list some selected and impressive works in recent time, for example L. Caffarelli et al [2], F. Duzaar et al [12,13,15], J.L. Vazquez et al [30,61,8,9] and the references therein.…”

Section: Introductionmentioning

confidence: 99%

On existence and regularity of a terminal value problem for the time fractional diffusion equation

et al. 2020

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In this paper we consider a final value problem for a diffusion equation with timespace fractional differentiation on a bounded domain D of R k , k ≥ 1, which includes the fractional power L β , 0 < β ≤ 1, of a symmetric uniformly elliptic operator L defined on L 2 (D). A representation of solutions is given by using the Laplace transform and the spectrum of L β . We establish some existence and regularity results for our problem in both the linear and nonlinear case.where ϕ is a given function. Here J is the interval (0, T ), The notation c D α t for 0 < α < 1 represents the left Caputo fractional derivative of order α which is defined by c D α t v(t) := I 1−α t (N.H. Tuan) Applied

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Existence and controllability of higher‐order nonlinear fractional integrodifferential systems via fractional resolvent

Haq

Sukavanam

2022

Math Methods in App Sciences

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This work analyzes the existence of solution and approximate controllability for higher‐order nonlinear fractional integrodifferential systems with Riemann–Liouville derivatives in Banach spaces. First, the definition of mild solution for the system is derived. Then a set of sufficient conditions for the existence of mild solution and approximate controllability of the system is obtained. The discussions are based on fixed point approach, and the theory of convolution and fractional resolvent. To illustrate the feasibility of developed theory, an example is given.

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Optimal existence and uniqueness theory for the fractional heat equation

Cited by 89 publications

References 45 publications

Asymptotic behavior of solutions to fractional diffusion–convection equations

Asymptotic behavior of solutions to fractional diffusion–convection equations

On existence and regularity of a terminal value problem for the time fractional diffusion equation

Existence and controllability of higher‐order nonlinear fractional integrodifferential systems via fractional resolvent

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