Page 670 - 8th European Congress of Mathematics ∙ 20-26 June 2021 ∙ Portorož, Slovenia ∙ Book of Abstracts
P. 670
TIAL DIFFERENTIAL EQUATIONS AND APPLICATIONS
Papers [2, 3] are devoted to investigation of the behavior of weak solutions in the case when the
blow-up set Ω0 ⊂ ∂Ω (LS-regime). The precise estimates of the limiting profile of solutions
were obtained, namely,
sup u(t, x) ψ(x), x ∈ Ω,
t→T
where function ψ is determined by characteristics of peaking of the boundary regime f .
As an application of these results we study the following parabolic quasilinear equation with
a nonlinear absorption term:
(|u|q−1u)t − ∆pu = −b(t, x)|u|λ−1u, (t, x) ∈ Q, λ > p q > 0, (3)
Here b(t, x) 0 is a degenerate absorption potential: b(t, x) → 0 as t → T ∀ x ∈ Ω. Precise
upper estimates for all weak solutions of equation (3) near to t = T (limiting profile of solution),
depending on the behavior of function b, were obtained in the papers [2, 4]. It is important to
underline that the obtained estimates don’t depend on initial and boundary values and hold for
large solutions of the equation (3) (if they exist).
References
[1] Galaktionov V.A., Shishkov A.E. Self-similar boundary blow-up for higher-order quasilin-
ear parabolic equations, Proc. Roy. Soc. Edinburgh. 135A (2005), 1195–1227.
[2] Shishkov A.E., Yevgenieva Ye.A. Localized peaking regimes for quasilinear parabolic
equations, Mathematische Nachrichten. 292 (2019), no. 6, 1349–1374.
[3] Shishkov A.E., Yevgenieva Ye.A. Localized blow-up regimes for quaislinear doubly degen-
erate parabolic equations, Mathematical Notes, 106 (2019), 639—650.
[4] Yevgenieva Ye.A. Propagation of singularities for large solutions of quasilinear parabolic
equations, Journal of Mathematical Physics, Analysis, Geometry. 15 (2019), no. 1, 131–
144.
Operator-norm asymptotics for thin elastic rods with rapidly oscillating
periodic properties
Josip Žubrinic´, josip.zubrinic@fer.hr
Faculty of Electrical Engineering and Computing, University of Zagreb, Croatia
Coauthors: Igor Velcˇic´, Kirill Cherednichenko
We provide norm-resolvent estimates in L2 → L2 and L2 → H1 operator norms, for the class of
problems in linear elasticity describing heterogeneous rods with rapidly oscillating coefficients
in the regime of moderate contrast.
The estimates are provided with respect to the period of material oscillations in the setting
of simultaneous homogenization and dimension reduction, while assuming that the period and
the rod thickness are of the same order. These estimates are expected to provide also sharp
estimates for the corresponding evolution problems.
The analysis is performed by the means of spectral analysis and Gelfand transform.
This is joint work with K. Cherednichenko and I. Velcˇic´.
668
Papers [2, 3] are devoted to investigation of the behavior of weak solutions in the case when the
blow-up set Ω0 ⊂ ∂Ω (LS-regime). The precise estimates of the limiting profile of solutions
were obtained, namely,
sup u(t, x) ψ(x), x ∈ Ω,
t→T
where function ψ is determined by characteristics of peaking of the boundary regime f .
As an application of these results we study the following parabolic quasilinear equation with
a nonlinear absorption term:
(|u|q−1u)t − ∆pu = −b(t, x)|u|λ−1u, (t, x) ∈ Q, λ > p q > 0, (3)
Here b(t, x) 0 is a degenerate absorption potential: b(t, x) → 0 as t → T ∀ x ∈ Ω. Precise
upper estimates for all weak solutions of equation (3) near to t = T (limiting profile of solution),
depending on the behavior of function b, were obtained in the papers [2, 4]. It is important to
underline that the obtained estimates don’t depend on initial and boundary values and hold for
large solutions of the equation (3) (if they exist).
References
[1] Galaktionov V.A., Shishkov A.E. Self-similar boundary blow-up for higher-order quasilin-
ear parabolic equations, Proc. Roy. Soc. Edinburgh. 135A (2005), 1195–1227.
[2] Shishkov A.E., Yevgenieva Ye.A. Localized peaking regimes for quasilinear parabolic
equations, Mathematische Nachrichten. 292 (2019), no. 6, 1349–1374.
[3] Shishkov A.E., Yevgenieva Ye.A. Localized blow-up regimes for quaislinear doubly degen-
erate parabolic equations, Mathematical Notes, 106 (2019), 639—650.
[4] Yevgenieva Ye.A. Propagation of singularities for large solutions of quasilinear parabolic
equations, Journal of Mathematical Physics, Analysis, Geometry. 15 (2019), no. 1, 131–
144.
Operator-norm asymptotics for thin elastic rods with rapidly oscillating
periodic properties
Josip Žubrinic´, josip.zubrinic@fer.hr
Faculty of Electrical Engineering and Computing, University of Zagreb, Croatia
Coauthors: Igor Velcˇic´, Kirill Cherednichenko
We provide norm-resolvent estimates in L2 → L2 and L2 → H1 operator norms, for the class of
problems in linear elasticity describing heterogeneous rods with rapidly oscillating coefficients
in the regime of moderate contrast.
The estimates are provided with respect to the period of material oscillations in the setting
of simultaneous homogenization and dimension reduction, while assuming that the period and
the rod thickness are of the same order. These estimates are expected to provide also sharp
estimates for the corresponding evolution problems.
The analysis is performed by the means of spectral analysis and Gelfand transform.
This is joint work with K. Cherednichenko and I. Velcˇic´.
668
