Andrey Shishkov
Doctor of Physics and Mathematics
Professor,
s.M. Nikol’skii Mathematical Institute
“I keep the subject constantly before me and wait until the first dawnings open slowly, by little and little, into a full and clear light” (Isaac Newton)
evolution of carriers of generalized solutions; behavior of solutions in unbounded domains
removable and unremovable singularities of solutions
equations of flow of thin films, Cahn-Hilliard equation
boundary modes with singular aggravation
localization of singularities of solutions
propagation of singularities of solutions
1972
Graduated from Faculty of Mechanics and Mathematics of Donetsk National University.
1974 - 1977
Postgraduate student at the Institute of Applied Mathematics and Mechanics of NAS of Ukraine.
1978
Candidate thesis on “Topological methods in the theory of nonlinear boundary value problems” was presented.
1977-1981
Junior researcher of the Institute of Applied Mathematics and Mechanics of NAS of Ukraine.
1981-1991
Senior researcher of the Institute of Applied Mathematics and Mechanics of NAS of Ukraine.
1990
Doctoral thesis on “Saint-Venant’s principle and its applications in the qualitative theory of nonlinear boundary value problems” was presented.
1991-2004
Leading researcher in the Department of partial differential equations of the Institute of applied mathematics and mechanics of NAS of Ukraine.
2004-2017
Head of the Department of partial differential equations of the Institute of Applied Mathematics and Mechanics of NAS of Ukraine.
2017 - present
Head of the Scientific Center of Nonlinear Problems of Mathematical Physics of S.M. Nikol’skii Mathematical Institute of RUDN University.
Teaching
A.E. Shishkov delivered lectures on the following topics at Donetsk National University:
- Partial differential equations;
- Ordinary differential equation;
- Functional analysis;
- Topological methods and functional analysis methods for nonlinear boundary value problems;
- Qualitative theory of nonlinear elliptic and parabolic equations.
Visiting scientist:
- 1987 - Mathematical Institute of Czechoslovak Academy of Sciences, Prague, Czechoslovakia;
- 1989 - Weierstass Institute, Berlin, Germany;
- 1994 - 1995 - Computer and Automation Institute (CAI) of Hungarian Academy of Sciences, Budapest, Hungary;
- 1995 – 1996 - Catania University, Italy;
- 1996 - Leiden University, the Netherlands; University of Besancon, France;
- 1997 - Instituto per le Applicazioni del Calcolo (IAC) “Mauro Picone”, Rome, Italy; University of Twente, the Netherlands; (University of Leiden, the Netherlands; Mathematical Institute of Wroclaw University, Poland;
- 1999 - University of Rome “La Sapienza”, Rome, Italy; University of Toulouse Paul Sabatier, Toulouse, France; University of Bath, England;
- 2001 – 2002 - University of Roma “La Sapienza”, University of Bath, UK; CAI of Hungarian Academy of Sciences, Budapest, Hungary; Israel Institute of Technology (Technion), Haifa, Israel.
- 2003 - University of Roma “La Sapienza”, University of Bath, UK.
- 2004 - University of Tours, France; University of Bath, UK.
- 2005 - Technion, Haifa, Israel; University of Bath, UK.
- January - February 2006 - University of Bath, UK.
- June - July 2006 - University of Tours, France; Humboldt University and Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Berlin, Germany.
- March – October 2007 - University of Tours, France; University of Bath, UK; University of Swansea, UK; University Technion, Haifa, Israel.
- September - December 2008 - University of Tours, France; Israel Institute of Technology (Technion), Haifa, Israel.
- October - November 2010, October 2011, November 2013, March 2015 - visiting professor, University of Tours, France.
- February - March 2011, January 2012, February 2013, November 2014, January 2018 - visiting professor Israel Institute of Technology (Technion), Haifa, Israel.
Member of the editorial boards of the journals:
- Ukrainian mathematical bulletin;
- Ukrainian mathematical journal;
- Abstract and Applied Analysis.
Science
- A number of variants of the method of local energy evaluation in the qualitative theory of nonlinear boundary value problems, which allowed to establish a number of fundamental results on the asymptotic and qualitative properties of generalized solutions of broad classes of quasi-linear elliptic, parabolic and some composite classes of differential equations was developed.
- Local variants of energy and entropy a priori evaluations of generalized solutions of the flow equations of thin viscous films, specific quasilinear parabolic equations of the fourth order were obtained in 1995. On the basis of these evaluations terms of finiteness of speed of distribution of carriers of the corresponding decisions were define in a number of works, and also the description of various scenarios of evolution of these carriers in time was given.
- A new energy method for studying linear parabolic boundary value problems with infinitely aggravating (growing) finite-time boundary value data was proposed in 1999. On the basis of this method, the exact classes of localized boundary modes (S-modes), that is, modes with exacerbation, generating solutions with spatially localized at the time of exacerbation of the singularity zone were described.
- Exact terms on the nature of the degeneracy capacity (Dini type terms) to ensure the absence of propagation of singularities of solutions and, consequently, the existence supersingular and “large” solutions were set together with professor L. Veron. It was proved with professor M. Marcus that on a wide class of potentials the Dini term is a necessary condition for the absence of singularity propagation along degeneracy varieties, that is, the criterion for the existence of these classes of singular solutions was found.
Scientific interests
- Qualitative theory of solutions of boundary value problems for quasilinear and nonlinear elliptic and parabolic equations of the second and high orders;
- Equation of flow of thin capillary films;
- Cahn-Hilliard equation;
- Theory of singular, super-singular and “large” solutions of nonlinear diffusion-absorption structure equations;
- Localized and non-localized boundary regimes with singular exacerbation.
We study the admissible growth at infinity of initial data of positive solutions of ∂tu−Δu+f(u)=0 in R+×RN when f(u) is a continuous function, mildly superlinear at infinity, the model case being f(u)=ulnα(1+u) with 1<α<2. We prove, in particular, that if the growth of the initial data at infinity is too strong, there is no more diffusion and the corresponding solution satisfies the ODE problem ∂tϕ+f(ϕ)=0 on R+ with ϕ(0)=∞.
We study equations of the form (*) ut - Δu + h(x)|u|q-1u = 0 in a half space R+N+1 . Here q > 1 and h is a continuous function in RN , vanishing at the origin and positive elsewhere. Let h ̅(s)=e^((-w(s))⁄s^2 ) and assume that (w(s))⁄s^(2 ) is monotone on (0, 1) and tends to infinity as s → 0. We show that, if w satisfies the Dini condition and h(x)≥h ̅(|x|)then there exists a maximal solution of (*). This solution tends to infinity as t → 0. On the contrary, if the Dini condition in the half space fails and h(x)≤h ̅(|x|), we construct a sequence of solutions whose initial data shrinks to the Dirac measure with infinite mass at the origin, but the limit of the sequence blows up everywhere on the positive time axis.
We study the long-time behavior of solutions of semi-linear parabolic equation of the following type ∂tu−Δu+a0(x)uq=0 where a0(x)⩾d0exp(−ω(|x|)/|x|2), d0>0, 1>q>0, and ω is a positive continuous radial function. We give a Dini-like condition on the function ω by two different methods which implies that any solution of the above equation vanishes in a finite time. The first one is a variant of a local energy method and the second one is derived from semi-classical limits of some Schrödinger operators.
We prove localization estimates for general 2mth-order quasilinear parabolic equations with boundary data blowing up in finite time, as t → T−. The analysis is based on energy estimates obtained from a system of functional inequalities expressing a version of Saint-Venant's principle from the theory of elasticity. We consider a special class of parabolic operators including those having fixed orders of algebraic homogenuity p > 0. This class includes the second-order heat equation and linear 2mth-order parabolic equations (p = 1), as well as many other higher-order quasilinear ones with p ≠ 1. Such homogeneous equations can be invariant under a group of scaling transformations, but the corresponding least-localized regional blow-up regimes are not group invariant and exhibit typical exponential singularities ~ e(T−t)−γ → ∞ as t → T−, with the optimal constant γ = 1/[m(p + 1) − 1] > 0. For some particular equations, we study the asymptotic blow-up behaviour described by perturbed first-order Hamilton–Jacobi equations, which shows that general estimates of exponential type are sharp.
In one space dimension, we study the finite speed of propagation property for zero contact-angle solutions of the thin-film equation in presence of a convective term. In the case of strong slippage, we obtain bounds in terms of the initial mass for both the "fast" and the "slow" interfaces, and for both short and (whenever the solution is global) large times, which we expect to be sharp. In the case of weak slippage, we obtain partial results for short times, which include a quantitative bound for moderate growths of the convective term. Our approach is based on energy/entropy methods shaped upon suitable extensions of Stampacchia's Lemma.
In this paper, we consider the Cauchy problem for a third-order partial differential model equation with a nonlinearity of the form |∇u|_q. It is proved that for q∈(1,3/2] there is no local weak solution to the Cauchy problem in R^3 for a sufficiently wide class of initial functions, while for q>3/2 there is a local weak solution.
The equation of slow diffusion with singular boundary data is considered. An estimate of all weak solutions of such a problem is obtained, provided that the boundary regime is localized. The comparative analysis of the results obtained by the method of energy estimates and the barrier technique for the equation of porous medium is presented.
Singular-sharpened regimes are studied for a wide class of second-order quasilinear parabolic equations. On the basis of energy methods, in a certain sense, accurate estimates of the final profile of the generalized solution in the vicinity of the exacerbation time are established, depending on the rate of increase in the global energy of this solution.
The mainstream is the study of the singular character of solutions to nonlinear elliptic and parabolic equations. We study the existence and properties of weak and entropy solutions to equations with weakly integrable and irregular data, the removability of singularities and the boundary regimes of singularly peaking.