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We study the chemotaxis system with singular sensitivity and logistic-type source: $u_{t} = Δ u - χ \nabla \cdot (u \nabla v / v) + r u - μ u^{k}$ , $0 = Δ v - v + u$ under the non-flux boundary conditions in a smooth bounded domain $Ω \subset ℝ^{n}$ , $χ, r, μ > 0$ , $k > 1$ and $n \geq 1$ . It is shown with $k \in (1, 2)$ that the system possesses a global generalized solution for $n \geq 2$ which is bounded when $χ > 0$ is suitably small related to $r > 0$ and the initial datum is properly small, and a global bounded classical solution for $n = 1$ .

Global Strichartz estimates for variable coefficient second order hyperbolic operators

Daniel Tataru (1999/2000)

Séminaire Équations aux dérivées partielles

Global time estimates for solutions to equations of dissipative type

Michael Ruzhansky, James Smith (2005)

Journées Équations aux dérivées partielles

Global time estimates of $L^{p} - L^{q}$ norms of solutions to general strictly hyperbolic partial differential equations are considered. The case of special interest in this paper are equations exhibiting the dissipative behaviour. Results are applied to discuss time decay estimates for Fokker-Planck equations and for wave type equations with negative mass.

Global well-posedness and blow up for the nonlinear fractional beam equations

Shouquan Ma, Guixiang Xu (2010)

Applicationes Mathematicae

We establish the Strichartz estimates for the linear fractional beam equations in Besov spaces. Using these estimates, we obtain global well-posedness for the subcritical and critical defocusing fractional beam equations. Of course, we need to assume small initial data for the critical case. In addition, by the convexity method, we show that blow up occurs for the focusing fractional beam equations with negative energy.

Global well-posedness and scattering for the defocusing $H^{\frac{1}{2}}$ -subcritical Hartree equation in $R^{d}$

Changxing Miao, Guixiang Xu, Lifeng Zhao (2009)

Annales de l'I.H.P. Analyse non linéaire

Global well-posedness and scattering for the derivative nonlinear Schrödinger equation with small rough data

Baoxiang Wang, Lijia Han, Chunyan Huang (2009)

Annales de l'I.H.P. Analyse non linéaire

Global well-posedness for the 2-D Boussinesq system with temperature-dependent thermal diffusivity

Qionglei Chen, Liya Jiang (2014)

Colloquium Mathematicae

We prove the global well-posedness of the 2-D Boussinesq system with temperature dependent thermal diffusivity and zero viscosity coefficient.

Global well-posedness of the initial value problem for the Hirota-Satsuma system.

Xueshang Feng (1994)

Manuscripta mathematica

Gradient estimates for a new class of degenerate elliptic and parabolic equations

Gary M. Lieberman (1994)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Gradient estimates for the p(x)-Laplacian equation in $ℝ^{N}$

Chao Zhang, Shulin Zhou, Bin Ge (2015)

Annales Polonici Mathematici

Under some assumptions on the function p(x), we obtain global gradient estimates for weak solutions of the p(x)-Laplacian type equation in $ℝ^{N}$ .

Gradient estimates in parabolic problems with unbounded coefficients

M. Bertoldi, S. Fornaro (2004)

Studia Mathematica

We study, with purely analytic tools, existence, uniqueness and gradient estimates of the solutions to the Neumann problems associated with a second order elliptic operator with unbounded coefficients in spaces of continuous functions in an unbounded open set Ω in $ℝ^{N}$ .

Gradient estimation of a $p$ -harmonic map.

Wang, Bei, Ma, Li (2010)

Electronic Journal of Differential Equations (EJDE) [electronic only]

Gradient potential estimates

Giuseppe Mingione (2011)

Journal of the European Mathematical Society

Pointwise gradient bounds via Riesz potentials like those available for the Poisson equation actually hold for general quasilinear equations.

Gradient regularity via rearrangements for $p$ -Laplacian type elliptic boundary value problems

Andrea Cianchi, Vladimir G. Maz'ya (2014)

Journal of the European Mathematical Society

A sharp estimate for the decreasing rearrangement of the length of the gradient of solutions to a class of nonlinear Dirichlet and Neumann elliptic boundary value problems is established under weak regularity assumptions on the domain. As a consequence, the problem of gradient bounds in norms depending on global integrability properties is reduced to one-dimensional Hardy-type inequalities. Applications to gradient estimates in Lebesgue, Lorentz, Zygmund, and Orlicz spaces are presented.

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