Stochastic energy balance climate models with Legendre weighted diffusion and an additive cylindrical Wiener process forcing

Alcolea-Díaz, Gema; Díaz, Jesús Ildefonso Díaz

doi:10.3934/dcdss.2021165

Cited by 4 publications

(9 citation statements)

References 45 publications

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“…We make a change of variables which is U α (t, x) = u α (t, x) − A(t). Now the comparison principle can be applied (even if the right hand side is a time depending nonlinear term: see, e.g., [22]) and we get that U α (x, t) ≤ W α (x, t) , which implies the desired comparison.…”

mentioning

confidence: 88%

“…provided (22). More precisely Theorem 4 [20,1] Let Ω ⊂ R N , N > 1 be a bounded open set where ∂Ω satisfies an inner and outer sphere condition.…”

mentioning

confidence: 99%

“…More precisely Theorem 4 [20,1] Let Ω ⊂ R N , N > 1 be a bounded open set where ∂Ω satisfies an inner and outer sphere condition. Assume (5), ( 21) and (22). Then there exists a unique classical solution U Ω ∞ of (32) whose explosive boundary profile satisfies…”

mentioning

confidence: 99%

“…(see [1]). ✷ Remark 9 Among other illustrative choices satisfying ( 5), ( 21) and (22) studied in [1] we pick up the function g(s) = e s for which…”

mentioning

confidence: 99%

“…Thus, to justify the comparison principle we can argue in a similar manner to the case of some stochastic parabolic equations with an additive noise (see, e.g., [22]). We make a change of variables which is U α (t, x) = u α (t, x) − A(t).…”

mentioning

confidence: 99%

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Controlled boundary explosions: dynamics after blow-up for some semilinear problems with global controls

Casal¹,

Díaz²,

Díaz³

et al. 2022

Preprint

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The main goal of this paper is to show that the blow up phenomenon (the explosion of the L ∞ -norm) of the solutions of several classes of evolution problems can be controlled by means of suitable global controls α(t) (i.e. only dependent on time) in such a way that the corresponding solution be well defined (as element of L 1 loc (0, +∞ : X), for some functional space X) after the explosion time. We start by considering the case of an ordinary differential equation with a superlinear term and show that the controlled explosion property holds by using a delayed control (built through the solution of the problem and by generalizing the nonlinear variation of constants formula, due to V.M. Alekseev in 1961, to the case of neutral delayed equations (since the control is only in the space W −1,q′ loc (0, +∞ : R), for some q > 1). We apply those arguments to the case of an evolution semilinear problem in which the differential equation is a semilinear elliptic equation with a superlinear absorption and the boundary condition is dynamic and involves the forcing superlinear term giving rise to the blow up phenomenon. We prove that, under a suitable balance between the forcing and the absorption terms, the blow up takes place only on the boundary of the spatial domain which here is assumed to be a ball BR and for a constant as initial datum.

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mentioning

confidence: 88%

“…provided (22). More precisely Theorem 4 [20,1] Let Ω ⊂ R N , N > 1 be a bounded open set where ∂Ω satisfies an inner and outer sphere condition.…”

mentioning

confidence: 99%

mentioning

confidence: 99%

“…(see [1]). ✷ Remark 9 Among other illustrative choices satisfying ( 5), ( 21) and (22) studied in [1] we pick up the function g(s) = e s for which…”

mentioning

confidence: 99%

mentioning

confidence: 99%

See 3 more Smart Citations

Controlled boundary explosions: dynamics after blow-up for some semilinear problems with global controls

Casal¹,

Díaz²,

Díaz³

et al. 2022

Preprint

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Approximation of the Heaviside function by sigmoidal functions in reaction–diffusion equations

Sun,

Han,

Kloeden

2024

Communications in Nonlinear Science and Numerical Simulation

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Controlled boundary explosions: Dynamics after blow-up for some semilinear problems with global controls

Casal¹,

Alcolea-Díaz²,

Díaz³

et al. 2023

DCDS

Self Cite

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The main goal of this paper is to show that the blow up phenomenon (the explosion of the <inline-formula><tex-math id="M1">\begin{document}$ {{\rm{L}}}^{\infty } $\end{document}</tex-math></inline-formula>-norm) of the solutions of several classes of evolution problems can be controlled by means of suitable global controls <inline-formula><tex-math id="M2">\begin{document}$ \alpha (t) $\end{document}</tex-math></inline-formula> (<inline-formula><tex-math id="M3">\begin{document}$ i.e. $\end{document}</tex-math></inline-formula> only dependent on time) in such a way that the corresponding solution be well defined (as element of <inline-formula><tex-math id="M4">\begin{document}$ {{\rm{L}}}_{loc}^{1}(0,+\infty : {{\rm{X}}}) $\end{document}</tex-math></inline-formula>, for some functional space <inline-formula><tex-math id="M5">\begin{document}$ {{\rm{X}}} $\end{document}</tex-math></inline-formula>) after the explosion time. We start by considering the case of an ordinary differential equation with a superlinear term and show that the controlled explosion property holds by using a delayed control (built through the solution of the problem and by generalizing the nonlinear variation of constants formula, due to V. M. Alekseev in 1961, to the case of neutral delayed equations (since the control is only in the space <inline-formula><tex-math id="M6">\begin{document}$ {{\rm{W}}}_{loc}^{-1,q\prime }(0,+\infty : \mathbb{R} ) $\end{document}</tex-math></inline-formula>, for some <inline-formula><tex-math id="M7">\begin{document}$ q>1 $\end{document}</tex-math></inline-formula>)<inline-formula><tex-math id="M8">\begin{document}$ . $\end{document}</tex-math></inline-formula> We apply those arguments to the case of an evolution semilinear problem in which the differential equation is a semilinear elliptic equation with a superlinear absorption and the boundary condition is dynamic and involves a forcing superlinear term giving rise to the blow up phenomenon. We prove that, under a suitable balance between the forcing and the absorption terms, the blow up takes place only on the boundary of the spatial domain which here is assumed to be a ball <inline-formula><tex-math id="M9">\begin{document}$ {{\rm{B}}}_{ {{\rm{R}}}} $\end{document}</tex-math></inline-formula> and for a constant as initial datum.

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Stochastic energy balance climate models with Legendre weighted diffusion and an additive cylindrical Wiener process forcing

Cited by 4 publications

References 45 publications

Controlled boundary explosions: dynamics after blow-up for some semilinear problems with global controls

Controlled boundary explosions: dynamics after blow-up for some semilinear problems with global controls

Approximation of the Heaviside function by sigmoidal functions in reaction–diffusion equations

Controlled boundary explosions: Dynamics after blow-up for some semilinear problems with global controls

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