On semilinear -Laplace equation

Kogoj, Alessia E.; Lanconelli, Ermanno

doi:10.1016/j.na.2011.10.007

Cited by 104 publications

(76 citation statements)

References 11 publications

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“…However, for the ∆ λ -Laplacian the Poincaré inequality (P ) can also be directly verified as in A, and the Sobolev embeddings (S) were proved in [22] and [14]. If we apply Theorem 1 to the ∆ λ -Laplacian we recover our previous result in [23].…”

Section: Resultssupporting

confidence: 65%

Attractors met X-elliptic operators

Kogoj

Sonner

2014

Journal of Mathematical Analysis and Applications

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Abstract. We consider degenerate parabolic and damped hyperbolic equations involving an operator L, that is X-elliptic with respect to a family of locally Lipschitz continuous vector fields X = {X 1 , . . . , Xm}. The local wellposedness is established under subcritical growth restrictions on the nonlinearity f , which are determined by the geometry and functional setting naturally associated to the family of vector fields X. Assuming additionally that f is dissipative, the global existence of solutions follows, and we can characterize their longtime behavior using methods from the theory of infinite dimensional dynamical systems.

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Section: Resultssupporting

confidence: 65%

Attractors met X-elliptic operators

Kogoj

Sonner

2014

Journal of Mathematical Analysis and Applications

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“…We recall some embedding results in [14], see also [7] for more general results related to the function space S 1 0 ( ).…”

Section: Function Spaces and Operatormentioning

confidence: 99%

On the Global Attractor for a Semilinear Strongly Degenerate Parabolic Equation

Thao

2015

Acta Math Vietnam

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We prove the existence of a global attractor in S 2 0 ( )∩L 2p−2 ( ) for a semilinear strongly degenerate parabolic equation in a bounded domain with the homogeneous Dirichlet boundary condition, in which the nonlinearity satisfies a polynomial type condition of arbitrary order and the external force belongs to L 2 ( ). This global attractor is then shown to have a finite fractal dimension in L 2 ( ). We also study the existence and exponential stability of the unique stationary solution to the problem.

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“…This operator was introduced by Franchi and Lanconelli [10], and recently reconsidered in [11] under the additional assumption that the operator is homogeneous of degree two with respect to a group dilation in R N , in [11] they were named as λ -Laplacians by Kogoj and Lanconelli. The λ -operator contains many degenerate elliptic operators such as the Grushin-type operator…”

Section: Introductionmentioning

confidence: 99%

“…We refer the interested reader to [20] for some important properties of this operator. The existence, non-existence and regularity of nontrivial weak solutions to problem (1.2) have been proved in [11] when the nonlinearity f (x, u) is subcritical and satisfies the well-known (AR) condition; see also [12][13][14] for related results. The main goal of this paper is to study the existence of weak solutions to problem (1.2) without the (AR) condition imposed on the nonlinear terms of subcritical polynomial growth.…”

Section: Introductionmentioning

confidence: 99%