Nonexistence of solutions to a hyperbolic equation with a time fractional damping

Mokhtar Kirane; Nasser Eddine Tatar

doi:10.4171/ZAA/1281

Nonexistence of solutions to a hyperbolic equation with a time fractional damping

Mokhtar Kirane, Nasser Eddine Tatar

Mathematics

King Fahd University for Petroleum and Minerals (KFUPM)

Research output: Contribution to journal › Article › peer-review

32 Scopus citations

Abstract

We consider the nonlinear hyperbolic equation u_tt - Δu + D₊^αu = h(t,x)|u|^p posed in Q := (0, ∞) × ℝ^N, where D₊^αu, 0 < α < 1 is a time fractional derivative, with given initial position and velocity u(0, x) = u₀(X) and u_t(0, x) = u ₁(x). We find the Fujita's exponent which separates in terms of p, α and N, the case of global existence from the one of nonexistence of global solutions. Then, we establish sufficient conditions on u₁(x) and h(x,t) assuring non-existence of local solutions.

Original language	British English
Pages (from-to)	131-142
Number of pages	12
Journal	Zeitschrift fur Analysis und ihre Anwendung
Volume	25
Issue number	2
DOIs	https://doi.org/10.4171/ZAA/1281
State	Published - 2006

Keywords

Fractional damping
Non-existence
Nonlinear hyperbolic equations

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abstract = "We consider the nonlinear hyperbolic equation utt - Δu + D+αu = h(t,x)|u|p posed in Q := (0, ∞) × ℝN, where D+αu, 0 < α < 1 is a time fractional derivative, with given initial position and velocity u(0, x) = u0(X) and ut(0, x) = u 1(x). We find the Fujita's exponent which separates in terms of p, α and N, the case of global existence from the one of nonexistence of global solutions. Then, we establish sufficient conditions on u1(x) and h(x,t) assuring non-existence of local solutions.",

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year = "2006",

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T1 - Nonexistence of solutions to a hyperbolic equation with a time fractional damping

AU - Kirane, Mokhtar

AU - Tatar, Nasser Eddine

PY - 2006

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N2 - We consider the nonlinear hyperbolic equation utt - Δu + D+αu = h(t,x)|u|p posed in Q := (0, ∞) × ℝN, where D+αu, 0 < α < 1 is a time fractional derivative, with given initial position and velocity u(0, x) = u0(X) and ut(0, x) = u 1(x). We find the Fujita's exponent which separates in terms of p, α and N, the case of global existence from the one of nonexistence of global solutions. Then, we establish sufficient conditions on u1(x) and h(x,t) assuring non-existence of local solutions.

AB - We consider the nonlinear hyperbolic equation utt - Δu + D+αu = h(t,x)|u|p posed in Q := (0, ∞) × ℝN, where D+αu, 0 < α < 1 is a time fractional derivative, with given initial position and velocity u(0, x) = u0(X) and ut(0, x) = u 1(x). We find the Fujita's exponent which separates in terms of p, α and N, the case of global existence from the one of nonexistence of global solutions. Then, we establish sufficient conditions on u1(x) and h(x,t) assuring non-existence of local solutions.

KW - Fractional damping

KW - Non-existence

KW - Nonlinear hyperbolic equations

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EP - 142

JO - Zeitschrift fur Analysis und ihre Anwendung

JF - Zeitschrift fur Analysis und ihre Anwendung

IS - 2

ER -

Nonexistence of solutions to a hyperbolic equation with a time fractional damping

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