On the asymptotic behavior of a subcritical convection-diffusion equation with nonlocal diffusion

Cazacu, Cristian; Ignat, Liviu I.; Pazoto, Ademir F.

doi:10.1088/1361-6544/aa773a

Cited by 7 publications

(6 citation statements)

References 20 publications

(93 reference statements)

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“…Through the paper we will consider nonnegative solutions. The general case of changing sign solutions can be analyzed following the same arguments as in [, Section 6]. We emphasize that since the nonlinearity should be locally Lipschitz we should impose

q > 1

.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…This implies that inequality becomes

W false(t false) + s (W^{2} false(t false) + p^{- β - 1} false(t false) q false(t false) W false(t false)) ⩽ W false(t - s false) + C s^{2} .

Letting

s \to 0

we obtain that for a.e.

t > 0

W

satisfies

W^{'} false(t false) + W^{2} false(t false) + p^{- β - 1} false(t false) q false(t false) W false(t false) ⩽ 0 .

Now it follows using classical ODEs arguments (see for example [, p. 3136]) that

W

satisfies

max false{W (t), 0 false} ⩽ \frac{1}{t}, \forall t > 0 .

To finish the proof it remains to prove claim . To do that, we use representation with suitable

r = r_{n}

depending on

x_{n}

that will be specified latter.…”

Section: Existence Of Solutions and Main Propertiesmentioning

confidence: 93%

“…This is a quite different topic, since the nonlocal operator does not provide any regularity for the solution, as it happens in the fractional derivative case, and then other techniques must be used. When

q < 2

, the first author considers the model

u_{t} = J ★ u - u - {(f false(u false))}_{x}

in . The asymptotic behavior is given by the solution of .…”

Section: Preliminariesmentioning

confidence: 99%

“…Then

z \in C_{b}^{\infty} false((0, \infty) \times double-struckR false)

and

z_{t} + false(q - 1 false) z^{1 - \frac{1}{q - 1}} {(- Δ)}^{α / 2} false[z^{\frac{1}{q - 1}} false] + z z_{x} = 0 .

Let

w false(t, x false) = z_{x} false(t, x false)

. Then

w \in C_{b}^{\infty} false((0, \infty) \times double-struckR false)

and it verifies

w_{t} + w^{2} + z w_{x} + z^{- β - 1} A false(w, z false) = 0 .

We continue as in following some ideas from . Let us denote

W false(t false) = {trueprefixsup}_{x \in R} w false(t, x false)

.…”

Section: Existence Of Solutions and Main Propertiesmentioning

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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q > 1

.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

“…This implies that inequality becomes

W false(t false) + s (W^{2} false(t false) + p^{- β - 1} false(t false) q false(t false) W false(t false)) ⩽ W false(t - s false) + C s^{2} .

Letting

s \to 0

we obtain that for a.e.

t > 0

W

satisfies

W^{'} false(t false) + W^{2} false(t false) + p^{- β - 1} false(t false) q false(t false) W false(t false) ⩽ 0 .

Now it follows using classical ODEs arguments (see for example [, p. 3136]) that

W

satisfies

max false{W (t), 0 false} ⩽ \frac{1}{t}, \forall t > 0 .

To finish the proof it remains to prove claim . To do that, we use representation with suitable

r = r_{n}

depending on

x_{n}

that will be specified latter.…”

Section: Existence Of Solutions and Main Propertiesmentioning

confidence: 93%

q < 2

, the first author considers the model

u_{t} = J ★ u - u - {(f false(u false))}_{x}

in . The asymptotic behavior is given by the solution of .…”

Section: Preliminariesmentioning

confidence: 99%

“…Then

z \in C_{b}^{\infty} false((0, \infty) \times double-struckR false)

and

z_{t} + false(q - 1 false) z^{1 - \frac{1}{q - 1}} {(- Δ)}^{α / 2} false[z^{\frac{1}{q - 1}} false] + z z_{x} = 0 .

Let

w false(t, x false) = z_{x} false(t, x false)

. Then

w \in C_{b}^{\infty} false((0, \infty) \times double-struckR false)

and it verifies

w_{t} + w^{2} + z w_{x} + z^{- β - 1} A false(w, z false) = 0 .

We continue as in following some ideas from . Let us denote

W false(t false) = {trueprefixsup}_{x \in R} w false(t, x false)

.…”

Section: Existence Of Solutions and Main Propertiesmentioning

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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“…One of the central properties in the analysis of Burgers' equation is the so-called Oleinik's estimate u x ≤ 1/t. In the nonlocal setting there are few results in this direction under the assumption that the convection is local (|u| q−1 u) x , q ∈ (1, 2], see [13,3]. The existence of similar estimates, in local or nonlocal form, for the models where the convection is nonlocal is, as far the authors know, an open problem.…”

Section: Theorem 12 Assume the Conditions In Theorem 11 And In Addmentioning

confidence: 99%

Long-time behavior for a nonlocal convection diffusion equation

Ignat

2017

Journal of Mathematical Analysis and Applications

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Abstract. In this paper we consider a nonlocal viscous Burgers' equation and study the well-posedness and asymptotic behaviour of its solutions. We prove that under the smallness assumption on the initial data the solutions behave as the self similar profiles of the Burgers' equation with Dirac mass as the initial datum. The first term in the asymptotic expansion of the solutions is obtained by rescaling the solutions and proving the compactness of their trajectories.Mathematics Subject Classification 2010 : 76Rxx, 35B40, 45M05, 45G10.

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Asymptotic stability of the nonlocal diffusion equation with nonlocal delay

Tang,

Wu,

Yuan

et al. 2024

Math Methods in App Sciences

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This work focuses on the asymptotic stability of nonlocal diffusion equations in ‐dimensional space with nonlocal time‐delayed response term. To begin with, we prove and ‐decay estimates for the fundamental solution of the linear time‐delayed equation by Fourier transform. For the considered nonlocal diffusion equation, we show that if , then the solution converges globally to the trivial equilibrium time‐exponentially. If , then the solution converges globally to the trivial equilibrium time‐algebraically. Furthermore, it can be proved that when , the solution converges globally to the positive equilibrium time‐exponentially, and when , the solution converges globally to the positive equilibrium time‐algebraically. Here, , and are the coefficients of each term contained in the linear part of the nonlinear term . All convergence rates above are and ‐decay estimates. The comparison principle and low‐frequency and high‐frequency analyses are significantly effective in proofs. Finally, our theoretical results are supported by numerical simulations in different situations.

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On the asymptotic behavior of a subcritical convection-diffusion equation with nonlocal diffusion

Cited by 7 publications

References 20 publications

Asymptotic behavior of solutions to fractional diffusion–convection equations

Asymptotic behavior of solutions to fractional diffusion–convection equations

Long-time behavior for a nonlocal convection diffusion equation

Asymptotic stability of the nonlocal diffusion equation with nonlocal delay

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