Infinity Laplace equation with a gradient term and a nonlinear inhomogeneous term

2019 ◽  
Vol 49 (12) ◽  
pp. 1947
Author(s):  
Zhang Shuxian ◽  
Zheng Sining ◽  
Wang Wei
Keyword(s):  
Laplace Equation ◽  
Gradient Term ◽  
Inhomogeneous Term

scholarly journals A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation

2020 ◽  
Vol 18 (1) ◽  
pp. 1685-1697
Author(s):  
Zhenyu Zhao ◽  
Lei You ◽  
Zehong Meng
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Solution Process ◽  
Tikhonov Regularization Method ◽  
Hermite Expansion ◽  
The Cauchy Problem ◽  
Ill Posed

Abstract In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.

Solvability of the abstract evolution equations in L s -spaces with critical temporal weights

2021 ◽  
Vol 19 (1) ◽  
pp. 111-120
Author(s):  
Qinghua Zhang ◽  
Zhizhong Tan
Keyword(s):  
Cauchy Problem ◽  
Asymptotic Behavior ◽  
Evolution Equation ◽  
Nonlinear Operator ◽  
Evolution Equations ◽  
Bounded Function ◽  
Inhomogeneous Term ◽  
Linear Evolution ◽  
Abstract Evolution Equations ◽  
Linear Evolution Equation

Abstract This paper deals with the abstract evolution equations in L s {L}^{s} -spaces with critical temporal weights. First, embedding and interpolation properties of the critical L s {L}^{s} -spaces with different exponents s s are investigated, then solvability of the linear evolution equation, attached to which the inhomogeneous term f f and its average Φ f \Phi f both lie in an L 1 / s s {L}_{1\hspace{-0.08em}\text{/}\hspace{-0.08em}s}^{s} -space, is established. Based on these results, Cauchy problem of the semi-linear evolution equation is treated, where the nonlinear operator F ( t , u ) F\left(t,u) has a growth number ρ ≥ s + 1 \rho \ge s+1 , and its asymptotic behavior acts like α ( t ) / t \alpha \left(t)\hspace{-0.1em}\text{/}\hspace{-0.1em}t as t → 0 t\to 0 for some bounded function α ( t ) \alpha \left(t) like ( − log t ) − p {\left(-\log t)}^{-p} with 2 ≤ p < ∞ 2\le p\lt \infty .

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