scholarly journals Non-Lipschitz heterogeneous reaction with a p-Laplacian operator

2021 ◽  
Vol 7 (3) ◽  
pp. 3395-3417
Author(s):  
José L. Díaz ◽  
Keyword(s):  
Blow Up ◽  
Heterogeneous Reaction ◽  
Propagation Speed ◽  
Weak Sense ◽  
Regularity Of Solutions ◽  
Laplacian Operator ◽  
Positivity Condition ◽  
Definition Of ◽  
Regularity Results ◽  
Analytical Approaches

<abstract><p>The intention along this work is to provide analytical approaches for a degenerate parabolic equation formulated with a p-Laplacian operator and heterogeneous non-Lipschitz reaction. Firstly, some results are discussed and presented in relation with uniqueness, existence and regularity of solutions. Due to the degenerate diffusivity induced by the p-Laplacian operator (specially when $ \nabla u = 0 $, or close zero), solutions are studied in a weak sense upon definition of an appropriate test function. The p-Laplacian operator is positive for positive solutions. This positivity condition is employed to show the regularity results along propagation. Afterwards, profiles of solutions are explored specially to characterize the propagating front that exhibits the property known as finite propagation speed. Finally, conditions are shown to the loss of compact support and, hence, to the existence of blow up phenomena in finite time.</p></abstract>

scholarly journals Space-time fractional stochastic equations on regular bounded open domains

2016 ◽  
Vol 19 (5) ◽  
Author(s):  
Vo V. Anh ◽  
Nikolai N. Leonenko ◽  
María D. Ruiz-Medina
Keyword(s):  
Evolution Equations ◽  
Asymptotic Properties ◽  
Sufficient Conditions ◽  
Weak Sense ◽  
Laplacian Operator ◽  
Mean Square ◽  
Fractional Polynomial ◽  
The Mean ◽  
Negative Laplacian ◽  
Definition Of

AbstractFractional (in time and in space) evolution equations defined on Dirichlet regular bounded open domains, driven by fractional integrated in time Gaussian spatiotemporal white noise, are considered here. Sufficient conditions for the definition of a weak-sense Gaussian solution, in the mean-square sense, are derived. The temporal, spatial and spatiotemporal Hölder continuity, in the mean-square sense, of the formulated solution is obtained, under suitable conditions, from the asymptotic properties of the Mittag-Leffler function, and the asymptotic order of the eigenvalues of a fractional polynomial of the Dirichlet negative Laplacian operator on such bounded open domains.

scholarly journals The diminished base locus is not always closed

2014 ◽  
Vol 150 (10) ◽  
pp. 1729-1741 ◽  
Author(s):  
John Lesieutre
Keyword(s):  
Blow Up ◽  
Weak Sense ◽  
Prime Divisors ◽  
General Fiber ◽  
Base Locus ◽  
The Family

AbstractWe exhibit a pseudoeffective $\mathbb{R}$-divisor ${D}_{\lambda }$ on the blow-up of ${\mathbb{P}}^{3}$ at nine very general points which lies in the closed movable cone and has negative intersections with a set of curves whose union is Zariski dense. It follows that the diminished base locus ${\boldsymbol{B}}_{-}({D}_{\lambda })={\bigcup }_{A\,\text{ample}}\boldsymbol{B}({D}_{\lambda }+A)$ is not closed and that ${D}_{\lambda }$ does not admit a Zariski decomposition in even a very weak sense. By a similar method, we construct an $\mathbb{R}$-divisor on the family of blow-ups of ${\mathbb{P}}^{2}$ at ten distinct points, which is nef on a very general fiber but fails to be nef over countably many prime divisors in the base.

Analytical, Numerical and Experimental Study of the Vibro-Acoustic Behaviour of a Stiffened Shell Structure

1995 ◽  
Author(s):  
François Charron ◽  
Raymond Panneton ◽  
Yvan Champoux ◽  
J.-M. Guérin ◽  
Sylvain Boily
Keyword(s):  
Ritz Method ◽  
Scale Model ◽  
Acoustic Behaviour ◽  
Initial Correlation ◽  
Measuring Techniques ◽  
Depth Study ◽  
Proper Definition ◽  
Definition Of ◽  
Analytical Approaches ◽  
Airplane Fuselage

Abstract The main objectives of this study are a better understanding of the vibro-acoustic behaviour of an airplane fuselage type structure including stiffeners and a better comprehension of the measuring techniques and the modelization approaches for this type of problem. In order to meet the above objectives, three different models were developed. The first one is an experimental model where the measured accelerations and acoustic pressures are used as a reference for the validation of predicted results. The second model is based on a semi-analytical approach. This model is derived from variational and integral approaches and solved, approximately using a Rayleigh-Ritz method. Finally, the last model is based on the finite element method. Several iterations have been necessary before reaching an excellent agreement between all three approaches, especially regarding acoustic responses. From the initial correlation between the measured and predicted results, two major problems were identified. The first one is related to convergence problem associated with the semi-analytical model when stiffeners are incorporated in the model. The second problem is associated with the proper definition of the fluid-structure intermodal coupling in the numerical and analytical approaches. This paper will present the various approaches and models. Furthermore, the investigation on the previous problems will be discussed in detail. In conclusion, new modelization limitations were identified and new modelization criteria for the intermodal coupling were developed from the present study. These results will be used for an in-depth study on the vibro-acoustic behaviour of 1/3 scale model of airplane fuselage.

scholarly journals Remarks on the Generalized Fractional Laplacian Operator

Mathematics ◽  
2019 ◽  
Vol 7 (4) ◽  
pp. 320 ◽  
Author(s):  
Chenkuan Li ◽  
Changpin Li ◽  
Thomas Humphries ◽  
Hunter Plowman
Keyword(s):  
Differential Equations ◽  
Fractional Laplacian ◽  
Laplacian Operator ◽  
Generalized Convolution ◽  
Riesz Fractional Derivative ◽  
Derivative Operator ◽  
Delta Sequence ◽  
Remote Sites ◽  
Definition Of ◽  
Fractional Laplacian Operator

The fractional Laplacian, also known as the Riesz fractional derivative operator, describes an unusual diffusion process due to random displacements executed by jumpers that are able to walk to neighbouring or nearby sites, as well as perform excursions to remote sites by way of Lévy flights. The fractional Laplacian has many applications in the boundary behaviours of solutions to differential equations. The goal of this paper is to investigate the half-order Laplacian operator ( − Δ ) 1 2 in the distributional sense, based on the generalized convolution and Temple’s delta sequence. Several interesting examples related to the fractional Laplacian operator of order 1 / 2 are presented with applications to differential equations, some of which cannot be obtained in the classical sense by the standard definition of the fractional Laplacian via Fourier transform.

scholarly journals Existence of Solutions for Unbounded Elliptic Equations with Critical Natural Growth

2018 ◽  
Vol 2018 ◽  
pp. 1-7
Author(s):  
Aziz Bouhlal ◽  
Abderrahmane El Hachimi ◽  
Jaouad Igbida ◽  
El Mostafa Sadek ◽  
Hamad Talibi Alaoui
Keyword(s):  
Elliptic Problem ◽  
Lebesgue Space ◽  
Elliptic Equations ◽  
Existence Of Solutions ◽  
Regularity Of Solutions ◽  
Natural Growth ◽  
Existence And Regularity Results ◽  
Regularity Results

We investigate existence and regularity of solutions to unbounded elliptic problem whose simplest model is {-div[(1+uq)∇u]+u=γ∇u2/1+u1-q+f  in  Ω,  u=0  on  ∂Ω,}, where 0<q<1, γ>0 and f belongs to some appropriate Lebesgue space. We give assumptions on f with respect to q and γ to show the existence and regularity results for the solutions of previous equation.

On the Mathematical Definition of Entropy

1974 ◽  
Vol 29 (9) ◽  
pp. 1239-1243
Author(s):  
Bo-Sture Skagerstam
Keyword(s):  
Weak Sense ◽  
Canonical Distribution ◽  
Mathematical Definition ◽  
Definition Of

We discuss the Baron-Jauch definition of entropy and show that it is the unique answer to an entropy which has the properties of extensivity, positivity and continuity in a weak sense. As an application we also show how one easily can derive the canonical distribution from this definition of entropy using information theoretical arguments.

scholarly journals Well-posedness and regularity results for a dynamic Von Kármán plate

1995 ◽  
Vol 18 (2) ◽  
pp. 237-244
Author(s):  
M. E. Bradley
Keyword(s):  
Strong Solutions ◽  
Free Edge ◽  
Regularity Of Solutions ◽  
Well Posedness ◽  
Boundary Damping ◽  
Von Karman ◽  
Edge Conditions ◽  
Von Kármán Plate ◽  
Regularity Results ◽  
Von Kármán

We consider the problem of well-posedness and regularity of solutions for a dynamic von Kármán plate which is clamped along one portion of the boundary and which experiences boundary damping through “free edge” conditions on the remainder of the boundary. We prove the existence of unique strong solutions for this system

scholarly journals Singularity Analysis for a Class of Porous Medium Equation with Time-Dependent Coefficients

2016 ◽  
Vol 2016 ◽  
pp. 1-5 ◽  
Author(s):  
Anyin Xia ◽  
Xianxiang Pu ◽  
Shan Li
Keyword(s):  
Porous Medium ◽  
Global Regularity ◽  
Blow Up ◽  
Porous Medium Equation ◽  
Singularity Analysis ◽  
Dirichlet Boundary Conditions ◽  
Time Dependent ◽  
Medium Equation ◽  
Regularity Results ◽  
Homogeneous Dirichlet Boundary Conditions

This paper concerns the singularity and global regularity for the porous medium equation with time-dependent coefficients under homogeneous Dirichlet boundary conditions. Firstly, some global regularity results are established. Furthermore, we investigate the blow-up solution to the boundary value problem. The upper and lower estimates to the lifespan of the singular solution are also obtained here.

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