Global existence and blow-up analysis for some degenerate and quasilinear parabolic systems

Self-similar solutions of a cross-diffusion parabolic system with variable density: explicit estimates and asymptotic behaviour

“…The local existence and uniqueness of classical solution also proven in [3]. It is also shown that: (i) when min {a 1 , ..., a n } ≤ λ 1 then there exists global positive classical solution, and all positive classical solutions cannot blow up in finite time in the meaning of maximum norm; (ii) when min {a 1 , ..., a n } > λ 1 , and the initial datum (u 10 , ..., u n0 ) satisfies some assumptions, then the positive classical solution is unique and blows up in finite time, where λ 1 is the first eigenvalue of ∆ in Ω with homogeneous Dirichlet boundary conditions.…”

“…Non-divergent form equations and system of equations (1) are often used to describe various physical phenomena, such as the diffusive process for biological species, the resistive diffusion phenomena in force-free magnetic fields, curve shortening flow, spreading of infectious disease and so on, see for [2][3][4][5][6][7][8].…”

“…Previous studies [3] with positive solutions of some degenerate and quasilinear parabolic systems are given by:…”

Finite Speed of the Perturbation Distribution and Asymptotic Behavior of the Solutions of a Parabolic System not in Divergence Form

“…, , , Not in divergence form equations and system of equations are often used to describe various physical phenomena, such as the diffusive process for biological species, the resistive diffusion phenomena in force-free magnetic fields, curve shortening flow, spreading of infectious disease and so on, see for [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15].…”

“…It is observed by many authors [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][19][20][21] nonlinear equations of non-divergence form are a source of new nonlinear effects, such as finite speed of a perturbation of distribution, space localization, blow up, etc. In particular, nonlinear problems in the non-divergence form of the following form…”