Regularizing Effects of Homogeneous Evolution Equations

Author(s):  
Michael G. Crandall ◽  
Philippe Benilan
Author(s):  
Paul Alphonse

We study the partial Gelfand–Shilov regularizing effect and the exponential decay for the solutions to evolution equations associated with a class of accretive non-selfadjoint quadratic operators, which fail to be globally hypoelliptic on the whole phase space. By taking advantage of the associated Gevrey regularizing effects, we study the null-controllability of parabolic equations posed on the whole Euclidean space associated with this class of possibly non-globally hypoelliptic quadratic operators. We prove that these parabolic equations are null-controllable in any positive time from thick control subsets. This thickness property is known to be a necessary and sufficient condition for the null-controllability of the heat equation posed on the whole Euclidean space. Our result shows that this geometric condition turns out to be a sufficient one for the null-controllability of a large class of quadratic differential operators.


2015 ◽  
Vol 11 (3) ◽  
pp. 3134-3138 ◽  
Author(s):  
Mostafa Khater ◽  
Mahmoud A.E. Abdelrahman

In this work, an extended Jacobian elliptic function expansion method is pro-posed for constructing the exact solutions of nonlinear evolution equations. The validity and reliability of the method are tested by its applications to the Couple Boiti-Leon-Pempinelli System which plays an important role in mathematical physics.


2019 ◽  
Vol 9 (3) ◽  
pp. 344-352 ◽  
Author(s):  
L.I. Stefanovich ◽  
O.Y. Mazur ◽  
V.V. Sobolev

Introduction: Within the framework of the phenomenological theory of phase transitions of the second kind of Ginzburg-Landau, the kinetics of ordering of a rapidly quenched highly nonequilibrium domain structure is considered using the lithium tantalate and lithium niobate crystals as an example. Experimental: Using the statistical approach, evolution equations describing the formation of the domain structure under the influence of a high-frequency alternating electric field in the form of a standing wave were obtained. Numerical analysis has shown the possibility of forming thermodynamically stable mono- and polydomain structures. It turned out that the process of relaxation of the system to the state of thermodynamic equilibrium can proceed directly or with the formation of intermediate quasi-stationary polydomain asymmetric phases. Results: It is shown that the formation of Regular Domain Structures (RDS) is of a threshold character and occurs under the influence of an alternating electric field with an amplitude less than the critical value, whose value depends on the field frequency. The conditions for the formation of RDSs with a micrometer spatial scale were determined. Conclusion: As shown by numerical studies, the RDSs obtained retain their stability, i.e. do not disappear even after turning off the external electric field. Qualitative analysis using lithium niobate crystals as an example has shown the possibility of RDSs formation in high-frequency fields with small amplitude under resonance conditions


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