Existence for a doubly nonlinear Volterra equation

AbstractIn this paper, we are interested in $L^{\infty }$ L ∞ decay estimates of weak solutions for the doubly nonlinear parabolic equation and the degenerate evolution m-Laplacian equation not in the divergence form. By a modified Moser’s technique we obtain $L^{\infty }$ L ∞ decay estimates of weak solutiona.

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Self-exciting multifractional processes

Journal of Applied Probability ◽

10.1017/jpr.2020.88 ◽

2021 ◽

Vol 58 (1) ◽

pp. 22-41

Author(s):

Fabian A. Harang ◽

Marc Lagunas-Merino ◽

Salvador Ortiz-Latorre

Keyword(s):

Stochastic Process ◽

Brownian Motion ◽

Rate Of Convergence ◽

Existence And Uniqueness ◽

Volterra Equation ◽

The Self ◽

The Past ◽

Stochastic Volterra Equation ◽

Multifractional Brownian Motion ◽

Self Organizing

AbstractWe propose a new multifractional stochastic process which allows for self-exciting behavior, similar to what can be seen for example in earthquakes and other self-organizing phenomena. The process can be seen as an extension of a multifractional Brownian motion, where the Hurst function is dependent on the past of the process. We define this by means of a stochastic Volterra equation, and we prove existence and uniqueness of this equation, as well as giving bounds on the p-order moments, for all $p\geq1$. We show convergence of an Euler–Maruyama scheme for the process, and also give the rate of convergence, which is dependent on the self-exciting dynamics of the process. Moreover, we discuss various applications of this process, and give examples of different functions to model self-exciting behavior.

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Harnack’s inequality for doubly nonlinear equations of slow diffusion type

Calculus of Variations and Partial Differential Equations ◽

10.1007/s00526-021-02044-z ◽

2021 ◽

Vol 60 (6) ◽

Author(s):

Verena Bögelein ◽

Andreas Heran ◽

Leah Schätzler ◽

Thomas Singer

Keyword(s):

Vector Field ◽

Harnack Inequality ◽

Parabolic Equations ◽

Full Range ◽

Nonlinear Parabolic Equations ◽

Growth Conditions ◽

Diffusion Type ◽

Slow Diffusion ◽

Nonlinear Parabolic ◽

Doubly Nonlinear

AbstractIn this article we prove a Harnack inequality for non-negative weak solutions to doubly nonlinear parabolic equations of the form $$\begin{aligned} \partial _t u - {{\,\mathrm{div}\,}}{\mathbf {A}}(x,t,u,Du^m) = {{\,\mathrm{div}\,}}F, \end{aligned}$$ ∂ t u - div A ( x , t , u , D u m ) = div F , where the vector field $${\mathbf {A}}$$ A fulfills p-ellipticity and growth conditions. We treat the slow diffusion case in its full range, i.e. all exponents $$m > 0$$ m > 0 and $$p>1$$ p > 1 with $$m(p-1) > 1$$ m ( p - 1 ) > 1 are included in our considerations.

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Global existence for a semilinear parabolic Volterra equation

Mathematische Zeitschrift ◽

10.1007/bf02570816 ◽

1992 ◽

Vol 209 (1) ◽

pp. 17-26 ◽

Cited By ~ 41

Author(s):

Philippe Clément ◽

Jan Prüss

Keyword(s):

Global Existence ◽

Volterra Equation ◽

Semilinear Parabolic

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A New Method Tracing Load-Deflection Equilibrium Path of a Doubly Nonlinear Truss

Applied Mechanics and Materials ◽

10.4028/www.scientific.net/amm.166-169.68 ◽

2012 ◽

Vol 166-169 ◽

pp. 68-72

Author(s):

Shu Tang Liu ◽

Qi Liang Long

Keyword(s):

Matrix Equation ◽

Stiffness Matrix ◽

Large Scale ◽

New Method ◽

Truss Structure ◽

Equilibrium Path ◽

Nodal Coordinates ◽

Doubly Nonlinear ◽

Global Stiffness Matrix

A new method tracing the load-deflection equilibrium path of a truss with doubly nonlinearity is proposed. The total global stiffness matrix equation has been formulated in terms of nodal coordinates, iteration formulations has been written through adopting a single control coordinate, so that an new method tracing the load-deflection equilibrium path has been proposed. Analysis results of Star dome truss and Schwedeler dome truss have shown that the proposed method is stable numerically, quick in convergence, high in degree of accuracy and easy in use. The proposed method can be used for large-scale truss structure.

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