EECT
A note on global well-posedness and blow-up of some semilinear evolution equations
Tarek Saanouni
Evolution Equations & Control Theory 2015, 4(3): 355-372 doi: 10.3934/eect.2015.4.355
We investigate the initial value problems for some semilinear wave, heat and Schrödinger equations in two space dimensions, with exponential nonlinearities. Using the potential well method based on the concepts of invariant sets, we prove either global well-posedness or finite time blow-up.
keywords: ground state. global existence nonlinear heat equation blow-up Nonlinear wave equation nonlinear Schrödinger equation
CPAA
Global well-posedness of some high-order semilinear wave and Schrödinger type equations with exponential nonlinearity
Tarek Saanouni
Communications on Pure & Applied Analysis 2014, 13(1): 273-291 doi: 10.3934/cpaa.2014.13.273
Extending previous works [47, 27, 30], we consider in even space dimensions the initial value problems for some high-order semi-linear wave and Schrödinger type equations with exponential nonlinearity. We obtain global well-posedness in the energy space.
keywords: high-order Schrödinger type equation High-order wave type equation Moser-Trudinger inequality well-posedness. energy estimate
CPAA
Well-posedness issues for some critical coupled non-linear Klein-Gordon equations
Radhia Ghanmi Tarek Saanouni
Communications on Pure & Applied Analysis 2019, 18(2): 603-623 doi: 10.3934/cpaa.2019030

The initial value problem for some coupled non-linear wave equations is investigated. In the defocusing case, global well-posedness and ill-posedness results are obtained. In the focusing sign, the existence of global and non global solutions are discussed via the potential-well theory. Finally, strong instability of standing waves are established.

keywords: Non-linear Klein-Gordon system global well-posedness scattering blow-up instability

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