• Title/Summary/Keyword: parabolic equation

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EXTINCTION AND POSITIVITY OF SOLUTIONS FOR A CLASS OF SEMILINEAR PARABOLIC EQUATIONS WITH GRADIENT SOURCE TERMS

  • Yi, Su-Cheol
    • Journal of the Chungcheong Mathematical Society
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    • v.30 no.4
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    • pp.397-409
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    • 2017
  • In this paper, we investigated the extinction, positivity, and decay estimates of the solutions to the initial-boundary value problem of the semilinear parabolic equation with nonlinear gradient source and interior absorption terms by using the integral norm estimate method. We found that the decay estimates depend on the choices of initial data, coefficients and domain, and the first eigenvalue of the Laplacean operator with homogeneous Dirichlet boundary condition plays an important role in the proofs of main results.

LONG-TIME BEHAVIOR FOR SEMILINEAR DEGENERATE PARABOLIC EQUATIONS ON ℝN

  • Cung, The Anh;Le, Thi Thuy
    • Communications of the Korean Mathematical Society
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    • v.28 no.4
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    • pp.751-766
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    • 2013
  • We study the existence and long-time behavior of solutions to the following semilinear degenerate parabolic equation on $\mathbb{R}^N$: $$\frac{{\partial}u}{{\partial}t}-div({\sigma}(x){\nabla}u+{\lambda}u+f(u)=g(x)$$, under a new condition concerning a variable non-negative diffusivity ${\sigma}({\cdot})$. Some essential difficulty caused by the lack of compactness of Sobolev embeddings is overcome here by exploiting the tail-estimates method.

GLOBAL ATTRACTOR FOR A SEMILINEAR STRONGLY DEGENERATE PARABOLIC EQUATION WITH EXPONENTIAL NONLINEARITY IN UNBOUNDED DOMAINS

  • Tu, Nguyen Xuan
    • Communications of the Korean Mathematical Society
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    • v.37 no.2
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    • pp.423-443
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    • 2022
  • We study the existence and long-time behavior of weak solutions to a class of strongly degenerate semilinear parabolic equations with exponential nonlinearities on ℝN. To overcome some significant difficulty caused by the lack of compactness of the embeddings, the existence of a global attractor is proved by combining the tail estimates method and the asymptotic a priori estimate method.

THE CAUCHY PROBLEM FOR A DENGERATE PARABOLIC EQUATION WITH ABSORPTION

  • Lee, Jin-Ho
    • Bulletin of the Korean Mathematical Society
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    • v.37 no.2
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    • pp.303-316
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    • 2000
  • The Cauchy problem for degenerate parabolic equations with absorption is studied. We prove the existence of a fundamental solution. Also a Harnack type inequality is established and the existence and uniqueness of initial trace for nonnegative solutions is shown.

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Effects of photostrictive actuator and active control of flexible membrane structure

  • Gajbhiye, S.C.;Upadhyay, S.H.;Harsha, S.P.
    • Smart Structures and Systems
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    • v.14 no.2
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    • pp.71-83
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    • 2014
  • The purpose of this paper is to investigate the flexible structure of parabolic shell using photostrictive actuators. The analysis is made to know its dynamic behavior and light-induced control forces for coupled parabolic shell. The effects of an actuator location as well as membrane and bending components under the control action have been analyzed considering the approximate spherical model. The parabolic membrane shell accuracy is being mathematically approximated and validated comparing the light induced control forces using approximate equivalent spherical shell model. The parabolic shell with kapton smart material and photostrictive actuators has been used to formulate the governing equation in the transverse direction. The Kirchhoff-Love assumptions are used to obtain the governing equation of shell with actuator. The mechanical membrane forces and bending moments for parabolic thin shell with actuator is used to analyze the dynamic effect. The results show that membrane control action is much more significant than bending control action. Photostrictive actuators oriented along circumferential direction (actuator-2) can give better control effect than actuators placed along longitudinal direction (actuator-1). The slight difference is observed between spherical and parabolic shell for a surface with focal length to the diameter ratio of 1.00 or more than unity. Space applications often have the shape of parabolical shells or shell of revolution, due to their required focusing, aiming, or reflecting performance. The present approach is focused that photostrictive actuators can effectively control the vibration of parabolical membrane shell. Also, the actuator's location plays an important role in defining the control force.

Analytic Error Caused by the Inconsistency of the Approximation Order between the Non Local Boundary Condition and the Parabolic Governing Equation (포물선 지배 방정식과 비국소적 경계조건의 근사 차수 불일치에 의한 해석적 오차)

  • Lee Keun-Hwa;Seong Woo-Jae
    • The Journal of the Acoustical Society of Korea
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    • v.25 no.5
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    • pp.229-238
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    • 2006
  • This paper shows the analytic error caused by the inconsistency of the approximation order between the non local boundary condition (NLBC) and the parabolic governing equation. To obtain the analytic error, we first transform the NLBC to the half space domain using plane wave analysis. Then, the analytic error is derived on the boundary between the true numerical domain and the half space domain equivalent to the NLBC. The derived analytic error is physically expressed as the artificial reflection. We examine the characteristic of the analytic error for the grazing angle, the approximation order of the PE or the NLBC. Our main contribution is to present the analytic method of error estimation and the application limit for the high order parabolic equation and the NLBC.

OSCILLATIONS OF CERTAIN NONLINEAR DELAY PARABOLIC BOUNDARY VALUE PROBLEMS

  • Zhang, Liqin;Fu, Xilin
    • Journal of applied mathematics & informatics
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    • v.8 no.1
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    • pp.137-149
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    • 2001
  • In this paper we consider some nonlinear parabolic partial differential equations with distributed deviating arguments and establish sufficient conditions for the oscillation of some boundary value problems.