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Energy methods for abstract nonlinear Schrödinger equations
So far there seems to be no abstract formulations for nonlinear Schrodinger equations (NLS). In some sense Cazenave[2, Chapter 3] has given a guiding principle to replace the free SchrodingerExpand
Boundedness in quasilinear Keller–Segel systems of parabolic–parabolic type on non-convex bounded domains
Abstract This paper deals with the quasilinear fully parabolic Keller–Segel system { u t = ∇ ⋅ ( D ( u ) ∇ u ) − ∇ ⋅ ( S ( u ) ∇ v ) , x ∈ Ω , t > 0 , v t = Δ v − v + u , x ∈ Ω , t > 0 , underExpand
Global existence of weak solutions to quasilinear degenerate Keller–Segel systems of parabolic–parabolic type
Abstract This paper deals with the quasilinear degenerate Keller–Segel system (KS) of parabolic–parabolic type. The global existence of weak solutions to (KS) is established when q m + 2 N (m denotesExpand
Global existence of weak solutions to quasilinear degenerate Keller-Segel systems of parabolic-parabolic type with small data
This paper deals with the quasilinear degenerate Keller–Segel system (KS) of “parabolic–parabolic” type. The global existence of weak solutions to (KS) with small initial data is established whenExpand
Monotonicity Method Applied to the Complex Ginzburg–Landau and Related Equations
Abstract Global existence of unique strong solutions is established for the complex Ginzburg–Landau equation ∂tu − (λ + iα)Δu + (κ + iβ)|u|p − 1u − γu = 0,  where λ > 0, κ > 0, α, β, γ ∈  R , p ≥ 1,Expand
Stabilization in the logarithmic Keller–Segel system
Abstract The Keller–Segel system u t = D Δ u − D χ ∇ ⋅ ( u v ∇ v ) , x ∈ Ω , t > 0 , v t = D Δ v − v + u , x ∈ Ω , t > 0 , is considered in a bounded domain Ω ⊂ R n , n ≥ 2 , with smooth boundary,Expand
Stabilization in a chemotaxis model for tumor invasion
This paper deals with the chemotaxis system \[ \begin{cases} u_t=\Delta u - \nabla \cdot (u\nabla v), \qquad x\in \Omega, \ t>0, \\ v_t=\Delta v + wz, \qquad x\in \Omega, \ t>0, \\ Expand
Boundedness of solutions to parabolic–elliptic Keller–Segel systems with signal‐dependent sensitivity
This paper deals with the parabolic–elliptic Keller–Segel system with signal-dependent chemotactic sensitivity function, under homogeneous Neumann boundary conditions in a smoothExpand
Blow-up in finite or infinite time for quasilinear degenerate Keller-Segel systems of parabolic-parabolic type
This paper gives a blow-up result for the quasilinear degenerate Keller-Segel systems of parabolic-parabolic type. It is known that the system has a global solvability in the case where $q < m +Expand
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