Well-posedness of semilinear strongly damped wave equations with fractional diffusion operators and $C^0$ potentials on arbitrary bounded domains

In this article the strongly damped wave equation is considered and a local well posedness result is obtained in the product space . The space of initial conditions is chosen according to the energy functional, whereas the approach used in this article is based on the theory of analytic semigroups as well as interpolation and extrapolation spaces. This functional analytic framework allows local existence results to be proved in the case of critically growing nonlinearities, which improves the existing results.

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Random attractors for stochastic retarded strongly damped wave equations with additive noise on bounded domains

Open Mathematics ◽

10.1515/math-2019-0038 ◽

2019 ◽

Vol 17 (1) ◽

pp. 472-486

Author(s):

Xiaoyao Jia ◽

Xiaoquan Ding

Keyword(s):

Asymptotic Behavior ◽

Wave Equations ◽

Additive Noise ◽

Bounded Domains ◽

Damped Wave Equation ◽

Bounded Smooth Domain ◽

Strongly Damped Wave Equation ◽

Random Attractors ◽

Bounded Smooth ◽

Strongly Damped

AbstractIn this paper we study the asymptotic behavior for a class of stochastic retarded strongly damped wave equation with additive noise on a bounded smooth domain in ℝd. We get the existence of the random attractor for the random dynamical systems associated with the equation.

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Global well-posedness of a class of fourth-order strongly damped nonlinear wave equations

Opuscula Mathematica ◽

10.7494/opmath.2019.39.2.297 ◽

2019 ◽

Vol 39 (2) ◽

pp. 297

Author(s):

Yang Yanbing ◽

Md Salik Ahmed ◽

Qin Lanlan ◽

Xu Runzhang

Keyword(s):

Nonlinear Wave ◽

Fourth Order ◽

Wave Equations ◽

Nonlinear Wave Equations ◽

Well Posedness ◽

Strongly Damped

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Approximate Solutions of Time Fractional Diffusion Wave Models

Mathematics ◽

10.3390/math7100923 ◽

2019 ◽

Vol 7 (10) ◽

pp. 923 ◽

Cited By ~ 1

Author(s):

Abdul Ghafoor ◽

Sirajul Haq ◽

Manzoor Hussain ◽

Poom Kumam ◽

Muhammad Asif Jan

Keyword(s):

Finite Difference ◽

Wave Equations ◽

Quadrature Rule ◽

Approximate Solutions ◽

Fractional Diffusion ◽

Haar Wavelets ◽

Test Problems ◽

Algebraic Equations ◽

Diffusion Wave ◽

Wave Models

In this paper, a wavelet based collocation method is formulated for an approximate solution of (1 + 1)- and (1 + 2)-dimensional time fractional diffusion wave equations. The main objective of this study is to combine the finite difference method with Haar wavelets. One and two dimensional Haar wavelets are used for the discretization of a spatial operator while time fractional derivative is approximated using second order finite difference and quadrature rule. The scheme has an excellent feature that converts a time fractional partial differential equation to a system of algebraic equations which can be solved easily. The suggested technique is applied to solve some test problems. The obtained results have been compared with existing results in the literature. Also, the accuracy of the scheme has been checked by computing L 2 and L ∞ error norms. Computations validate that the proposed method produces good results, which are comparable with exact solutions and those presented before.

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On a time fractional diffusion with nonlocal in time conditions

Advances in Difference Equations ◽

10.1186/s13662-021-03365-1 ◽

2021 ◽

Vol 2021 (1) ◽

Author(s):

Nguyen Hoang Tuan ◽

Nguyen Anh Triet ◽

Nguyen Hoang Luc ◽

Nguyen Duc Phuong

Keyword(s):

Fourier Series ◽

Diffusion Equation ◽

Convergence Rate ◽

Exact Solutions ◽

Mild Solution ◽

Integral Condition ◽

Fractional Diffusion ◽

Well Posedness ◽

Nonlocal Integral Condition ◽

Regularized Solution

AbstractIn this work, we consider a fractional diffusion equation with nonlocal integral condition. We give a form of the mild solution under the expression of Fourier series which contains some Mittag-Leffler functions. We present two new results. Firstly, we show the well-posedness and regularity for our problem. Secondly, we show the ill-posedness of our problem in the sense of Hadamard. Using the Fourier truncation method, we construct a regularized solution and present the convergence rate between the regularized and exact solutions.

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