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Journal articles on the topic 'Singular quadratic gradient term'

Author: Grafiati

Published: 22 February 2023

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1

Kumar, D., and J. Tyagi. "Singular elliptic equations with quadratic gradient term." Journal of Mathematical Analysis and Applications 502, no. 1 (2021): 125245. http://dx.doi.org/10.1016/j.jmaa.2021.125245.

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2

Porru, Giovanni, and Antonio Vitolo. "Problems for elliptic singular equations with a quadratic gradient term." Journal of Mathematical Analysis and Applications 334, no. 1 (2007): 467–86. http://dx.doi.org/10.1016/j.jmaa.2006.12.017.

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3

Zhou, Wenshu, Xiaodan Wei, and Xulong Qin. "Nonexistence of solutions for singular elliptic equations with a quadratic gradient term." Nonlinear Analysis: Theory, Methods & Applications 75, no. 15 (2012): 5845–50. http://dx.doi.org/10.1016/j.na.2012.06.001.

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4

Baraket, Sami, Imen Bazarbacha, Saber Kharrati, and Taieb Ouni. "Singular Limits for 2-Dimensional Elliptic Problem with Exponentially Dominated Nonlinearity and a Quadratic Convection Term." Analele Universitatii "Ovidius" Constanta - Seria Matematica 21, no. 1 (2013): 19–50. http://dx.doi.org/10.2478/auom-2013-0002.

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Abstract We study existence of solutions with singular limits for a two-dimensional semilinear elliptic problem with exponential dominated nonlinearity and a quadratic convection non linear gradient term, imposing Dirichlet boundary condition. This paper extends previous results obtained in [1], [3], [4] and some references therein for related issues.

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5

Giachetti, Daniela, Francesco Petitta, and Sergio Segura de León. "Elliptic equations having a singular quadratic gradient term and a changing sign datum." Communications on Pure and Applied Analysis 11, no. 5 (2012): 1875–95. http://dx.doi.org/10.3934/cpaa.2012.11.1875.

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6

Boccardo, Lucio. "Dirichlet problems with singular and gradient quadratic lower order terms." ESAIM: Control, Optimisation and Calculus of Variations 14, no. 3 (2008): 411–26. http://dx.doi.org/10.1051/cocv:2008031.

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7

Croce, Gisella. "An elliptic problem with degenerate coercivity and a singular quadratic gradient lower order term." Discrete & Continuous Dynamical Systems - S 5, no. 3 (2012): 507–30. http://dx.doi.org/10.3934/dcdss.2012.5.507.

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8

Benrhouma, Mohamed. "On a singular elliptic system with quadratic growth in the gradient." Journal of Mathematical Analysis and Applications 448, no. 2 (2017): 1120–46. http://dx.doi.org/10.1016/j.jmaa.2016.11.038.

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9

Arcoya, David, Sara Barile, and Pedro J. Martínez-Aparicio. "Singular quasilinear equations with quadratic growth in the gradient without sign condition." Journal of Mathematical Analysis and Applications 350, no. 1 (2009): 401–8. http://dx.doi.org/10.1016/j.jmaa.2008.09.073.

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10

Giarrusso, Ester, and Giovanni Porru. "Problems for elliptic singular equations with a gradient term." Nonlinear Analysis: Theory, Methods & Applications 65, no. 1 (2006): 107–28. http://dx.doi.org/10.1016/j.na.2005.08.007.

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11

Shi, Peihu, and Mingxin Wang. "Self-similar singular solution of doubly singular parabolic equation with gradient absorption term." Journal of Inequalities and Applications 2006 (2006): 1–21. http://dx.doi.org/10.1155/jia/2006/20438.

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12

Mosco, Umberto, and Maria Agostina Vivaldi. "An Example of Fractal Singular Homogenization." gmj 14, no. 1 (2007): 169–93. http://dx.doi.org/10.1515/gmj.2007.169.

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Abstract We construct a sequence of quadratic weighted energy forms in an open domain of the plane, that 𝑀-converges to an energy form with a singular fractal term. The weights belong to the Muckenoupt class 𝐴2 and have pointwise singularities. The result implies the spectral convergence of a sequence of second-order weighted elliptic operators in divergence form in the plane to a singular elliptic operator with a second order fractal term.

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13

Zhang, Wen, Jian Zhang, and Heilong Mi. "Ground states and multiple solutions for Hamiltonian elliptic system with gradient term." Advances in Nonlinear Analysis 10, no. 1 (2020): 331–52. http://dx.doi.org/10.1515/anona-2020-0113.

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Abstract This paper is concerned with the following nonlinear Hamiltonian elliptic system with gradient term $$\begin{array}{} \displaystyle \left\{\,\, \begin{array}{ll} -{\it\Delta} u +\vec{b}(x)\cdot \nabla u+V(x)u = H_{v}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N},\\[-0.3em] -{\it\Delta} v -\vec{b}(x)\cdot \nabla v +V(x)v = H_{u}(x,u,v)\,\,\hbox{in}\,\mathbb{R}^{N}.\\ \end{array} \right. \end{array}$$ Compared with some existing issues, the most interesting feature of this paper is that we assume that the nonlinearity satisfies a local super-quadratic condition, which is weaker than the usual global super-quadratic condition. This case allows the nonlinearity to be super-quadratic on some domains and asymptotically quadratic on other domains. Furthermore, by using variational method, we obtain new existence results of ground state solutions and infinitely many geometrically distinct solutions under local super-quadratic condition. Since we are without more global information on the nonlinearity, in the proofs we apply a perturbation approach and some special techniques.

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14

Giachetti, Daniela, and Sergio Segura de León. "Quasilinear stationary problems with a quadratic gradient term having singularities." Journal of the London Mathematical Society 86, no. 2 (2012): 585–606. http://dx.doi.org/10.1112/jlms/jds014.

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15

Abdellaoui, B., A. Attar, and S. E. Miri. "Nonlinear singular elliptic problem with gradient term and general datum." Journal of Mathematical Analysis and Applications 409, no. 1 (2014): 362–77. http://dx.doi.org/10.1016/j.jmaa.2013.07.017.

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16

Ivanova, I. D. "Null shells and double layers in quadratic gravity." Journal of Physics: Conference Series 2081, no. 1 (2021): 012020. http://dx.doi.org/10.1088/1742-6596/2081/1/012020.

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Abstract For a singular hypersurface of arbitrary type in quadratic gravity motion equations were obtained using only the least action principle. It turned out that the coefficients in the motion equations are zeroed with a combination corresponding to the Gauss-Bonnet term. Therefore it does not create neither double layers nor thin shells. It has been demonstrated that there is no “external pressure” for any type of null singular hypersurface. It turned out that null spherically symmetric singular hupersurfaces in quadratic gravity cannot be a double layer, and only thin shells are possible. The system of motion equations in this case is reduced to one which is expressed through the invariants of spherical geometry along with the Lichnerowicz conditions. Spherically symmetric null thin shells were investigated for spherically symmetric solutions of conformal gravity as applications, in particular, for various vacua and Vaidya-type solutions.

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17

TANG, LEI-HAN. "PHASON ELASTICITY IN EQUILIBRIUM QUASICRYSTAL MODELS." Modern Physics Letters B 03, no. 15 (1989): 1121–26. http://dx.doi.org/10.1142/s0217984989001722.

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The phason elasticity of random tiling and quasiperiodic crystal models is discussed. Long-wavelength phasons in the random tiling model are described by a square-gradient free energy of entropic origin. In the quasiperiodic crystal model, thermal fluctuations lead to a transition from the zero temperature singular linear phason elastic energy to a quadratic phason elastic energy at high temperatures.

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18

Arcoya, David, and Lourdes Moreno-Mérida. "The effect of a singular term in a quadratic quasi-linear problem." Journal of Fixed Point Theory and Applications 19, no. 1 (2016): 815–31. http://dx.doi.org/10.1007/s11784-016-0374-0.

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19

Ghergu, Marius, and Vicenţiu Rădulescu. "Bifurcation for a class of singular elliptic problems with quadratic convection term." Comptes Rendus Mathematique 338, no. 11 (2004): 831–36. http://dx.doi.org/10.1016/j.crma.2004.03.020.

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20

Carmona, José, and Tommaso Leonori. "A uniqueness result for a singular elliptic equation with gradient term." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 148, no. 5 (2018): 983–94. http://dx.doi.org/10.1017/s0308210518000112.

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We prove the uniqueness of a solution for a problem whose simplest model iswith k ≥ 1, 0 f ∈ L∞(Ω) and Ω is a bounded domain of ℝN, N ≥ 2. So far, uniqueness results are known for k < 1, while existence holds for any k ≥ 1 and f positive in open sets compactly embedded in a neighbourhood of the boundary. We extend the uniqueness results to the k ≥ 1 case and show, with an example, that existence does not hold if f is zero near the boundary. We even deal with the uniqueness result when f is replaced by a nonlinear term λuq with 0 < q < 1 and λ > 0.

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21

POLLOCK, M. D. "SUPERSYMMETRY AND DIMENSIONALITY IN THE SUPERSTRING THEORY." International Journal of Modern Physics A 24, no. 23 (2009): 4373–88. http://dx.doi.org/10.1142/s0217751x09045832.

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The realization of non-linear global supersymmetry in the superstring theory requires the quadratic fermionic Lagrangian [Formula: see text], defined from the D-dimensional, Minkowski-space energy–momentum tensor Tmn, to have the same form as the quadratic gravitational contribution [Formula: see text] to the superstring Lagrangian. Here, we prove that this condition is only satisfied for the heterotic string theory after reduction to D = 4, irrespective of whether the original source of [Formula: see text] in ten or twenty-six dimensions is the quadratic term [Formula: see text] or the quartic term [Formula: see text]. If [Formula: see text] derives from [Formula: see text], the solution is D = 4 (or the unphysical value D = 1), while if we suppose that D≠4 and [Formula: see text] dominates, we obtain the (singular) solution (D-2)3 = 0. The world sheet is also discussed. The bosonic string and type-II superstring, on the other hand, yield solutions for D which are complex, non-integral, or at the singular point D = 2, where the Einstein equations hold identically.

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22

Arcoya, David, and Sergio Segura de León. "Uniqueness of solutions for some elliptic equations with a quadratic gradient term." ESAIM: Control, Optimisation and Calculus of Variations 16, no. 2 (2008): 327–36. http://dx.doi.org/10.1051/cocv:2008072.

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23

Braeuning, Sebastian, Tom A. Jelmert, and Sven A. Vik. "The effect of the quadratic gradient term on variable-rate well-tests." Journal of Petroleum Science and Engineering 21, no. 3-4 (1998): 203–22. http://dx.doi.org/10.1016/s0920-4105(98)00073-4.

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24

Xu-long, Cao, Tong Deng-ke, and Wang Rui-he. "Exact solutions for nonlinear transient flow model including a quadratic gradient term." Applied Mathematics and Mechanics 25, no. 1 (2004): 102–9. http://dx.doi.org/10.1007/bf02437298.

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25

Fernández, Luis A. "Controllability properties for some semilinear parabolic PDE with a quadratic gradient term." Applied Mathematics Letters 25, no. 12 (2012): 2184–87. http://dx.doi.org/10.1016/j.aml.2012.05.019.

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26

Dall'Aglio, A., D. Giachetti, C. Leone, and S. Segura de León. "Quasi-linear parabolic equations with degenerate coercivity having a quadratic gradient term." Annales de l'Institut Henri Poincare (C) Non Linear Analysis 23, no. 1 (2006): 97–126. http://dx.doi.org/10.1016/j.anihpc.2005.02.006.

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27

Yao, Shengwei, Yuping Wu, Jielan Yang, and Jieqiong Xu. "A Three-Term Gradient Descent Method with Subspace Techniques." Mathematical Problems in Engineering 2021 (January 7, 2021): 1–7. http://dx.doi.org/10.1155/2021/8867309.

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We proposed a three-term gradient descent method that can be well applied to address the optimization problems in this article. The search direction of the obtained method is generated in a specific subspace. Specifically, a quadratic approximation model is applied in the process of generating the search direction. In order to reduce the amount of calculation and make the best use of the existing information, the subspace was made up of the gradient of the current and prior iteration point and the previous search direction. By using the subspace-based optimization technology, the global convergence result is established under Wolfe line search. The results of numerical experiments show that the new method is effective and robust.

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28

Yao, Shengwei, Yuping Wu, Jielan Yang, and Jieqiong Xu. "A Three-Term Gradient Descent Method with Subspace Techniques." Mathematical Problems in Engineering 2021 (January 7, 2021): 1–7. http://dx.doi.org/10.1155/2021/8867309.

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We proposed a three-term gradient descent method that can be well applied to address the optimization problems in this article. The search direction of the obtained method is generated in a specific subspace. Specifically, a quadratic approximation model is applied in the process of generating the search direction. In order to reduce the amount of calculation and make the best use of the existing information, the subspace was made up of the gradient of the current and prior iteration point and the previous search direction. By using the subspace-based optimization technology, the global convergence result is established under Wolfe line search. The results of numerical experiments show that the new method is effective and robust.

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29

DIRAR, M., ALI EL-TAHIR, and M. H. SHADDAD. "A GENERALIZED FIELD COSMOLOGY." Modern Physics Letters A 13, no. 37 (1998): 3025–31. http://dx.doi.org/10.1142/s0217732398003211.

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The generalized field equation is used to construct a cosmological mode by suggesting a Lagrangian which consists of a quadratic term besides the linear one. Solutions of this model are non-singular and do not possess the horizon, entropy, flatness or the age problem.

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30

Cuccu, F., E. Giarrusso, and G. Porru. "Boundary behaviour for solutions of elliptic singular equations with a gradient term." Nonlinear Analysis: Theory, Methods & Applications 69, no. 12 (2008): 4550–66. http://dx.doi.org/10.1016/j.na.2007.11.011.

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31

Shang, Haifeng. "On the Cauchy problem for the singular parabolic equations with gradient term." Journal of Mathematical Analysis and Applications 378, no. 2 (2011): 578–91. http://dx.doi.org/10.1016/j.jmaa.2011.01.024.

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32

Zouatini, M. A., F. Mokhtari, and H. Khelifi. "Degenerate elliptic problem with singular gradient lower order term and variable exponents." Mathematical Modeling and Computing 10, no. 1 (2023): 133–46. http://dx.doi.org/10.23939/mmc2023.01.133.

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In this paper, we prove the existence and regularity of weak solutions for a class of nonlinear elliptic equations with degenerate coercivity and singular lower-order terms with natural growth with respect to the gradient and Lm(⋅) (m(x)≥1) data. The functional setting involves Lebesgue–Sobolev spaces with variable exponents.

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33

Li, Jiang-Tao, Ren-Shi Nie, Yong-Lu Jia, and Dan-Ling Wang. "Semianalytical Solution of the Nonlinear Dual-Porosity Flow Model with the Quadratic Pressure Gradient Term." Mathematical Problems in Engineering 2015 (2015): 1–8. http://dx.doi.org/10.1155/2015/387368.

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The nonlinear dual-porosity flow model, specifically considering the quadratic pressure gradient term, wellbore storage coefficient, well skin factor, and interporosity flow of matrix to natural fractures, was established for well production in a naturally fractured formation and then solved using a semianalytical method, including Laplace transform and a transformation of the pressure function. Analytical solution of the model in Laplace space was converted to numerical solution in real space using Stehfest numerical inversion. Nonlinear flow process for well production in a naturally fractured formation with different external boundaries was simulated and analyzed using standard pressure curves. Influence of the quadratic pressure gradient coefficient on pressure curves was studied qualitatively and quantitatively in conditions of a group of fixed model parameters. The research results show that the semianalytical modelling method is applicable in simulating the nonlinear dual-porosity flow behavior.

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34

Coclite, Mario Michele. "Two perturbation results for nondegenerate solutions of some semilinear Dirichlet problems." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 111, no. 1-2 (1989): 1–11. http://dx.doi.org/10.1017/s0308210500024963.

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SynopsisThe stability of nondegenerate solutions of some semilinear Dirichlet problems is studied. Two specific situations are considered: firstly, a singular perturbation of the differential operator; secondly, a perturbation of the nonlinear term using a term which also depends on the gradient of the solution.

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35

Chu, Chong-Sun, and Hai Siong Tan. "Generalized Darmois–Israel Junction Conditions." Universe 8, no. 5 (2022): 250. http://dx.doi.org/10.3390/universe8050250.

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We present a general method to derive the appropriate Darmois–Israel junction conditions for gravitational theories with higher-order derivative terms by integrating the bulk equations of motion across the singular hypersurface. In higher-derivative theories, the field equations can contain terms which are more singular than the Dirac delta distribution. To handle them appropriately, we formulate a regularization procedure based on representing the delta function as the limit of a sequence of classical functions. This procedure involves imposing suitable constraints on the extrinsic curvature such that the field equations are compatible with the singular source being a delta distribution. As explicit examples of our approach, we demonstrate in detail how to obtain the generalized junction conditions for quadratic gravity, F(R) theories, a 4D low-energy effective action in string theory, and action terms that are Euler densities. Our results are novel, and refine the accuracy of previously claimed results in F(R) theories and quadratic gravity. In particular, when the coupling constants of quadratic gravity are those for the Gauss–Bonnet case, our junction conditions reduce to the known ones for the latter obtained independently by boundary variation of a surface term in the action. Finally, we briefly discuss a couple of applications to thin-shell wormholes and stellar models.

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36

Wu, Gaoyi, Yong Li, and Gonglin Yuan. "A Three-Term Conjugate Gradient Algorithm with Quadratic Convergence for Unconstrained Optimization Problems." Mathematical Problems in Engineering 2018 (June 27, 2018): 1–15. http://dx.doi.org/10.1155/2018/4813030.

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This paper further studies the WYL conjugate gradient (CG) formula with βkWYL≥0 and presents a three-term WYL CG algorithm, which has the sufficiently descent property without any conditions. The global convergence and the linear convergence are proved; moreover the n-step quadratic convergence with a restart strategy is established if the initial step length is appropriately chosen. Numerical experiments for large-scale problems including the normal unconstrained optimization problems and the engineer problems (Benchmark Problems) show that the new algorithm is competitive with the other similar CG algorithms.

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37

Fujita, Yasuhiro, and Paola Loreti. "Long-time behavior of solutions to Hamilton–Jacobi equations with quadratic gradient term." Nonlinear Differential Equations and Applications NoDEA 16, no. 6 (2009): 771–91. http://dx.doi.org/10.1007/s00030-009-0034-9.

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38

Rezende, Manuela C., and Carlos Alberto Santos. "Positive solutions for a singular and superlinear $p$-Laplacian problem with gradient term." Rocky Mountain Journal of Mathematics 49, no. 6 (2019): 2029–46. http://dx.doi.org/10.1216/rmj-2019-49-6-2029.

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39

Haitao, Yang. "A compact support principle for singular elliptic differential inequalities with a gradient term." Nonlinear Analysis: Theory, Methods & Applications 67, no. 7 (2007): 2157–66. http://dx.doi.org/10.1016/j.na.2006.08.041.

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40

Dinu, Teodora-Liliana. "Entire positive solutions of the singular Emden-Fowler equation with nonlinear gradient term." Results in Mathematics 43, no. 1-2 (2003): 96–100. http://dx.doi.org/10.1007/bf03322725.

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41

BARAKET, SAMI, and TAIEB OUNI. "SINGULAR LIMITS FOR 2-DIMENSIONAL ELLIPTIC PROBLEMS INVOLVING EXPONENTIAL NONLINEARITIES WITH SUB-QUADRATIC CONVECTION TERM." Glasgow Mathematical Journal 55, no. 3 (2013): 537–57. http://dx.doi.org/10.1017/s0017089512000729.

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AbstractLet Ω be a bounded domain with smooth boundary in ℝ2, q∈[1,2) and x1, x2,. . .,xm ∈ Ω. In this paper we are concerned with the following type of problem: \[ -\Delta u-\lambda|\nabla u|^q = \rho^{2}e^{u}, \] with u = 0 on ∂ Ω. We use some nonlinear domain decomposition method to construct a positive weak solution vρ,λ in Ω, which tends to a singular function at each xi as the parameters ρ and λ tend to 0 independently.

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42

Munteanu, Ionuţ, and Michael Röckner. "Total variation flow perturbed by gradient linear multiplicative noise." Infinite Dimensional Analysis, Quantum Probability and Related Topics 21, no. 01 (2018): 1850003. http://dx.doi.org/10.1142/s0219025718500030.

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We consider stochastic nonlinear diffusion equations with a highly singular diffusivity term and multiplicative gradient-type noise. We study existence and uniqueness of non-negative variational solutions in terms of stochastic variational inequalities. We also show the positivity preserving property of the solutions and extinction in finite time with probability one. These kinds of equations arise, e.g., in the use for simulation of image restoring techniques or for modeling turbulence.

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43

Ren, Junjie, Qiao Zheng, and Chunlan Zhao. "A Modified Blasingame Production Analysis Method for Vertical Wells Considering the Quadratic Gradient Term." Energies 12, no. 11 (2019): 2092. http://dx.doi.org/10.3390/en12112092.

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Fluid flow in actual oil reservoirs is consistent with material balance, which should be described by the nonlinear governing equation, including the quadratic gradient term (QGT). Nonetheless, the widely-used Blasingame production decline analysis (BPDA) is established based on the conventional governing equation neglecting the QGT, which leads to some errors in the interpretation of production data under some conditions, such as wells producing at a large drawdown pressure. This work extends BPDA to incorporate the effect of the QGT by modifying material balance time and normalized rate functions. The step-by-step procedure for the proposed production decline analysis (PPDA) is presented and compared with that for BPDA. The simulated cases for various production scenarios are used to validate PPDA. A field case is employed to show the applicability of PPDA in practice. Comparisons between the results obtained by BPDA and PPDA are analyzed in detail. It is found that BPDA overestimates the permeability and original oil-in-place, while PPDA works well. Compared with BPDA, PPDA can be employed to obtain more accurate original oil-in-place and reservoir properties, especially when wells produce at a large drawdown pressure.

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44

Ren, Junjie, and Ping Guo. "Analytical method for transient flow rate with the effect of the quadratic gradient term." Journal of Petroleum Science and Engineering 162 (March 2018): 774–84. http://dx.doi.org/10.1016/j.petrol.2017.11.005.

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45

Li, Wei, Xiao-ping Li, Shun-chu Li, and Quan-yong Li. "The Similar Structure of Solutions In Fractal Multilayer Reservoir Including A Quadratic Gradient Term." Journal of Hydrodynamics 24, no. 3 (2012): 332–38. http://dx.doi.org/10.1016/s1001-6058(11)60252-7.

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46

Chakrabarty, Chayan, S. M. Farouq Ali, and W. S. Tortike. "Analytical solutions for radial pressure distribution including the effects of the quadratic-gradient term." Water Resources Research 29, no. 4 (1993): 1171–77. http://dx.doi.org/10.1029/92wr02892.

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47

Boccardo, Lucio. "The impact of a “quadratic gradient” term in a system of Schrödinger–Maxwell equations." Probability, Uncertainty and Quantitative Risk 7, no. 4 (2022): 275–82. http://dx.doi.org/10.3934/puqr.2022016.

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48

Atifi, Khalid, Idriss Boutaayamou, Hamed Ould Sidi, and Jawad Salhi. "An Inverse Source Problem for Singular Parabolic Equations with Interior Degeneracy." Abstract and Applied Analysis 2018 (December 9, 2018): 1–16. http://dx.doi.org/10.1155/2018/2067304.

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The main purpose of this work is to study an inverse source problem for degenerate/singular parabolic equations with degeneracy and singularity occurring in the interior of the spatial domain. Using Carleman estimates, we prove a Lipschitz stability estimate for the source term provided that additional measurement data are given on a suitable interior subdomain. For the numerical solution, the reconstruction is formulated as a minimization problem using the output least squares approach with the Tikhonov regularization. The Fréchet differentiability of the Tikhonov functional and the Lipschitz continuity of the Fréchet gradient are proved. These properties allow us to apply gradient methods for numerical solution of the considered inverse source problem.

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49

Dao, Nguyen Anh, Jesus Ildefonso Díaz, and Quan Ba Hong Nguyen. "Pointwise Gradient Estimates in Multi-dimensional Slow Diffusion Equations with a Singular Quenching Term." Advanced Nonlinear Studies 20, no. 2 (2020): 477–502. http://dx.doi.org/10.1515/ans-2020-2076.

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AbstractWe consider the high-dimensional equation {\partial_{t}u-\Delta u^{m}+u^{-\beta}{\chi_{\{u>0\}}}=0}, extending the mathematical treatment made in 1992 by B. Kawohl and R. Kersner for the one-dimensional case. Besides the existence of a very weak solution {u\in\mathcal{C}([0,T];L_{\delta}^{1}(\Omega))}, with {u^{-\beta}\chi_{\{u>0\}}\in L^{1}((0,T)\times\Omega)}, {\delta(x)=d(x,\partial\Omega)}, we prove some pointwise gradient estimates for a certain range of the dimension N, {m\geq 1} and {\beta\in(0,m)}, mainly when the absorption dominates over the diffusion ({1\leq m<2+\beta}). In particular, a new kind of universal gradient estimate is proved when {m+\beta\leq 2}. Several qualitative properties (such as the finite time quenching phenomena and the finite speed of propagation) and the study of the Cauchy problem are also considered.

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50

Khelifi, Hichem. "Existence and Regularity for Solution to a Degenerate Problem with Singular Gradient Lower Order Term." Moroccan Journal of Pure and Applied Analysis 8, no. 3 (2022): 310–27. http://dx.doi.org/10.2478/mjpaa-2022-0022.

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Abstract We study the existence and regularity results for non-linear elliptic equation with degenerate coercivity and a singular gradient lower order term. The model problems is { - d i v ( b ( x ) | ∇ u | p - 2 ∇ u ( 1 + | u | ) γ ) + | ∇ u | p | u | θ = f , i n Ω , u = 0 , o n ∂ Ω , \left\{ {\matrix{ { - div\left( {b\left( x \right){{{{\left| {\nabla u} \right|}^{p - 2}}\nabla u} \over {\left( {1 + \left| u \right|} \right)\gamma }}} \right) + {{{{\left| {\nabla u} \right|}^p}} \over {{{\left| u \right|}^\theta }}} = f,} \hfill & {in\,\Omega ,} \hfill \cr {u = 0,} \hfill & {on\,\partial \Omega ,} \hfill \cr } } \right. swhere Ω is a bounded open subset in ℝN, 1 ≤ θ < 2, p > 2 and γ > 0. We will show that, even if the lower order term is singular, we obtain existence and regularity of positive solution, under various assumptions on the summability of the source f.

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