Limiting grow-up behavior for a one-parameter family of dissipative PDEs

Bruschi, Simone M.; Carvalho, Alexandre de; Pimentel, Juliana

doi:10.1512/iumj.2020.69.7836

Cited by 2 publications

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“…The profiles of equilibria at infinity are compactified profiles of metasolutions through (1.5). For the approximation of metasolutions by solutions, see [26], which can be compared to the approximation scheme in [12] using our current Poincaré projection approach. The unbounded solutions consist of the projected grow-up solutions (thick grey arrow), H up , which are heteroclinics from projected bounded equilibria towards equilibra at infinity, E ∞ .…”

Section: Resultsmentioning

confidence: 99%

Unbounded Sturm attractors for quasilinear parabolic equations

Lappicy,

Fernandes

2024

Proceedings of the Edinburgh Mathematical Society

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We analyse the asymptotic dynamics of quasilinear parabolic equations when solutions may grow up (i.e. blow up in infinite time). For such models, there is a global attractor which is unbounded and the semiflow induces a nonlinear dynamics at infinity by means of a Poincaré projection. In case the dynamics at infinity is given by a semilinear equation, then it is gradient, consisting of the so-called equilibria at infinity and their corresponding heteroclinics. Moreover, the diffusion and reaction compete for the dimensionality of the induced dynamics at infinity. If the equilibria are hyperbolic, we explicitly prove the occurrence of heteroclinics between bounded equilibria and/or equilibria at infinity. These unbounded global attractors describe the space of admissible initial data at event horizons of certain black holes.

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Section: Resultsmentioning

confidence: 99%

Unbounded Sturm attractors for quasilinear parabolic equations

Lappicy,

Fernandes

2024

Proceedings of the Edinburgh Mathematical Society

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“…The set I is the maximal invariant set if I = {x ∈ X : there is a global solution γ : R → X through x which is bounded in the past} It is also possible to define another notion, which coincides with the classical concept in the theory of global attractors. Note that in Example 1 I \ I b consists of heteroclinics to infinity as defined by [6], see also [18] and the articles [3,13,21,27].…”

Section: Definitionmentioning

confidence: 99%

“…Finally, in Sect. 2.7, we discus, on an abstract level, the dynamics at infinity, which corresponds to the equilibria at infinity and their connections in [3,6,18,21,27], and we prove, using the Hopf lemma, and homotopy invariance of the degree, that the attractor at infinity always coincides with the whole unit sphere in E + .…”

Section: Introduction and Summary Of Conceptsmentioning

confidence: 95%

“…The next series of our results is inspired by the works of [6,18] later extended and refined in [3,21,27]. Namely, for the case of the semigroup governed by the following PDE in one space dimension u t = u x x + bu + f (u, u x , x), on the space interval, with appropriate (typically Neumann) boundary data, and sufficiently large constant b > 0 the authors there study the detailed structure of the unbounded attractor which consists of equilibria, their heteroclinic connections, appropriately understood 'equilibria at infinity' and their connections, as well as heteroclinic connections from the equilibria to 'equilibria at infinity'.…”

Section: Introduction and Summary Of Conceptsmentioning

confidence: 99%

“…such that the orbits are not absorbed by one given bounded set, and they possibly diverge to infinity when time tends to infinity, but there is no blow-up in finite time. We remark, that in recent years, stemming from [9], there appeared significant work in the framework of unbounded attractors, cf., [3,6,13,18,21,27]. An abstract approach to autonomous and non-autonomous unbounded attractors has been very recently proposed in [4].…”

Section: Introduction and Summary Of Conceptsmentioning

confidence: 99%

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Autonomous and Non-autonomous Unbounded Attractors in Evolutionary Problems

et al. 2022

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If the semigroup is slowly non-dissipative, i.e., its black solutions can diverge to infinity as time tends to infinity, one still can study its dynamics via the approach by the unbounded attractors—the counterpart of the classical notion of global attractors. We continue the development of this theory started by Chepyzhov and Goritskii (Unbounded attractors of evolution equations. Advances in Soviet mathematics, American Mathematical Society, Providence, 1992). We provide the abstract results on the unbounded attractor existence, and we study the properties of these attractors, as well as of unbounded $$\omega $$ ω -limit sets in slowly non-dissipative setting. We also develop the pullback non-autonomous counterpart of the unbounded attractor theory. The abstract theory that we develop is illustrated by the analysis of the autonomous problem governed by the equation $$u_t = Au + f(u)$$ u t = A u + f ( u ) . In particular, using the inertial manifold approach, we provide the criteria under which the unbounded attractor coincides with the graph of the Lipschitz function, or becomes close to the graph of the Lipschitz function for large argument.

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Limiting grow-up behavior for a one-parameter family of dissipative PDEs

Cited by 2 publications

References 0 publications

Unbounded Sturm attractors for quasilinear parabolic equations

Unbounded Sturm attractors for quasilinear parabolic equations

Autonomous and Non-autonomous Unbounded Attractors in Evolutionary Problems

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