DSpace at Texas State University
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The University Scholarship digital repository system captures, stores, indexes, preserves, and distributes digital research material.2023-03-30T18:31:11ZExistence of global weak solutions for a p-Laplacian inequality with strong dissipation in noncylindrical domains
https://digital.library.txstate.edu/handle/10877/16513
Existence of global weak solutions for a p-Laplacian inequality with strong dissipation in noncylindrical domains
Ferreira, Jorge; Piskin, Erhan; Shahrouzi, Mohammad; Cordeiro, Sebastiao; Raposo, Carlos Alberto
In this work, we obtain global solutions for nonlinear inequalities of p-Laplacian type in noncylindrical domains, for the unilateral problem with strong dissipation uʺ - Δpu - Δu' - ƒ ≥ 0 in Q0, where Δp is the nonlinear p-Laplacian operator with 2 ≤ p < ∞, and Q0 is the noncylindrical domain. Our proof is based on a penalty argument by J. L. Lions and Faedo-Galerkin approximations.
2022-01-27T00:00:00ZBlow-up for parabolic equations in nonlinear divergence form with time-dependent coefficients
https://digital.library.txstate.edu/handle/10877/16512
Blow-up for parabolic equations in nonlinear divergence form with time-dependent coefficients
Shen, Xuhui; Ding, Juntang
In this article, we study the blow-up of solutions to the nonlinear parabolic equation in divergence form, (h(u))t = n∑i,j=1 (ɑ ij(x)uxi)xj - k(t)ƒ(u) in Ω x (0, t*), n∑i,j=1 ɑ ij(x)uxi vj = g(u) on ∂Ω x (0, t*), u(x, 0) = u0(x) ≥ 0 in Ω̅, where Ω is a bounded convex domain in ℝn (n ≥ 2) with smooth boundary ∂Ω. By constructing suitable auxiliary functions and using a differential inequality technique, when Ω ⊂ ℝn (n ≥ 2), we establish conditions for the solution blow up at a finite time, and conditions for the solution to exist for all time. Also, we find an upper bound for the blow-up time. In addition, when Ω ⊂ ℝn with (n ≥ 3), we use a Sobolev inequality to obtain a lower bound for the blow-up time.
2022-01-25T00:00:00ZAsymptotic behavior of solutions to 3D Kelvin-Voigt-Brinkman-Forchheimer equations with unbounded delays
https://digital.library.txstate.edu/handle/10877/16511
Asymptotic behavior of solutions to 3D Kelvin-Voigt-Brinkman-Forchheimer equations with unbounded delays
Thuy, Le Thi
In this article we consider a 3D Kelvin Voigt Brinkman Forchheimer equations involving unbounded delays in a bounded domain Ω ⊂ ℝ3. First, we show the existence and uniqueness of weak solutions by using the Galerkin approximations method and the energy method. Second, we prove the existence and uniqueness of stationary solutions by employing the Brouwer fixed point theorem. Finally, we study the stability of stationary solutions via the direct classical approach and the construction of a Lyapunov function. We also give a sufficient condition for the polynomial stability of the stationary solution for a special case of unbounded variable delay.
2022-01-17T00:00:00ZMathematical analysis of a Dupuit-Richards model
https://digital.library.txstate.edu/handle/10877/16510
Mathematical analysis of a Dupuit-Richards model
Al Nazer, Safaa; Rosier, Carole; Tsegmid, Munkhgerel
This article concerns an alternative model to the 3D-Richards equation to describe the flow of water in shallow aquifers. The model couples the two dominant types of flow existing in the aquifer. The first is described by the classic Richards problem in the upper capillary fringe. The second results from Dupuit's approximation after vertical integration of the conservation laws between the bottom of the aquifer and the saturation interface. The final model consists of a strongly coupled system of parabolic-type partial differential equations that are defined in a time-dependent domain. First, we show how taking the low compressibility of the fluid into account eliminates the nonlinearity in the time derivative of the Richards equation. Then, the general framework of parabolic equations is used in non-cylindrical domains to give a global in time existence result to this problem.
2022-01-17T00:00:00Z