First and second critical exponents for an inhomogeneous damped wave equation with mixed nonlinearities

Bessem Samet; Bessem Samet

doi:10.3934/math.2020452

AIMS Mathematics

2020, Volume 5, Issue 6: 7055-7070. doi: 10.3934/math.2020452

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First and second critical exponents for an inhomogeneous damped wave equation with mixed nonlinearities

Bessem Samet ^,

Department of Mathematics, College of Science, King Saud University, P.O. Box 2455, Riyadh, 11451, Saudi Arabia

Received: 25 July 2020 Accepted: 31 August 2020 Published: 09 September 2020
MSC : 35L05, 35B44, 35B33

We investigate the Cauchy problem for the nonlinear damped wave equation $u_{tt}-\Delta u +u_t = |u|^p+|\nabla u|^q +w(x)$, where $N\geq 1$, $p, q>1$, $w\in L^1_{loc}(\mathbb{R}^N)$, $w\geq 0$ and $w\not\equiv 0$. Namely, we first obtain the Fujita critical exponent for the considered problem. Next, we determine its second critical exponent in the sense of Lee and Ni. In particular, we show that the nonlinear gradient term $|\nabla u|^q$ induces a phenomenon of discontinuity of the Fujita critical exponent.
- damped wave equation,
- mixed nonlinearity,
- global weak solution,
- critical exponent
Citation: Bessem Samet. First and second critical exponents for an inhomogeneous damped wave equation with mixed nonlinearities[J]. AIMS Mathematics, 2020, 5(6): 7055-7070. doi: 10.3934/math.2020452

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Abstract

We investigate the Cauchy problem for the nonlinear damped wave equation $u_{tt}-\Delta u +u_t = |u|^p+|\nabla u|^q +w(x)$, where $N\geq 1$, $p, q>1$, $w\in L^1_{loc}(\mathbb{R}^N)$, $w\geq 0$ and $w\not\equiv 0$. Namely, we first obtain the Fujita critical exponent for the considered problem. Next, we determine its second critical exponent in the sense of Lee and Ni. In particular, we show that the nonlinear gradient term $|\nabla u|^q$ induces a phenomenon of discontinuity of the Fujita critical exponent.

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