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Global solutions of semilinear parabolic equations are studied in the case when some weak a priori estimate for solutions of the problem under consideration is already known. The focus is on the rapid growth of the nonlinear term for which existence of the semigroup and certain dynamic properties of the considered system can be justified. Examples including the famous Cahn-Hilliard equation are finally discussed.

Global solutions with infinite energy for the one-dimensional Zakharov system.

Pecher, Hartmut (2005)

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

Global solvability for certain classes of underdetermined systems of vector fields.

A.P. Bergamasco, P.D. Cordaro, G. Petronilho (1996)

Mathematische Zeitschrift

Global solvability for the degenerate Kirchhoff equation

Fumihiko Hirosawa (1998)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

Global solvability in the parabolic-elliptic chemotaxis system with singular sensitivity and logistic source

Xiangdong Zhao (2024)

Czechoslovak Mathematical Journal

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 solvability of a mixed problem for a nonlinear hyperbolic-parabolic equation in noncylindrical domains.

Ferreira, J., Lar'kin, N.A. (1996)

Portugaliae Mathematica

Global solvability of the mixed problem for first order functional partial differential equations

Jan Turo (1991)

Annales Polonici Mathematici

Global solvability to the Kirchhoff equation for a new class of initial data.

Manfrin, R. (2002)

Portugaliae Mathematica. Nova Série

Global sound waves for quasilinear second order wave equations.

Paul Godin (1994)

Mathematische Annalen

Global sovability for the degenerate Krichhoff equation with real anlytic data.

P. D'Ancona, S. Spagnolo (1992)

Inventiones mathematicae

Global special regular solutions to the Navier-Stokes equations in axially symmetric domains under boundary slip conditions [Book]

Wojciech M. Zajączkowski (2005)

Global stability of travelling fronts for a damped wave equation with bistable nonlinearity

Thierry Gallay, Romain Joly (2009)

Annales scientifiques de l'École Normale Supérieure

We consider the damped wave equation $α u_{t t} + u_{t} = u_{x x} - V^{'} (u)$ on the whole real line, where $V$ is a bistable potential. This equation has travelling front solutions of the form $u (x, t) = h (x - s t)$ which describe a moving interface between two different steady states of the system, one of which being the global minimum of $V$ . We show that, if the initial data are sufficiently close to the profile of a front for large $| x |$ , the solution of the damped wave equation converges uniformly on $ℝ$ to a travelling front as $t \to + \infty$ . The proof of this global stability...

Global Strichartz estimates for variable coefficient second order hyperbolic operators

Daniel Tataru (1999/2000)

Séminaire Équations aux dérivées partielles

Global strong solution and its decay properties for the Navier-Stokes equations in three dimensional domains with non-compact boundaries.

Hideo Kozono, Takayoshi Ogawa (1994)

Mathematische Zeitschrift

Global strong solution of the Navier-Stokes equations in 4 and 5 dimensional unbounded domains

Hideo Kozono, Hermann Sohr (1999)

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

Global strong solutions in Sobolev or Lebesgue spaces to the incompressible Navier-Stokes equations in $ℝ^{3}$

F. Planchon (1996)

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

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