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Papers by Sawanya Sakuntasathien

Thesis (M.Sc.)--Chulalongkorn University, 2001 As a general tool for integration, the Lebesgue in... more Thesis (M.Sc.)--Chulalongkorn University, 2001 As a general tool for integration, the Lebesgue integral is very useful. Its definition is quite general and it has a well-developed theory. However, it does have its limitations. An important limitation is that a function will be Lebesgue integrable only if its absolute value has finite integral, and there exist simple examples of functions that do not satisfy this property, yet intuition suggests they should be integrable. The generalized Riemann integral has helped to solve this problem. Unfortunately, it has a useful theory only for integration over subsets of finite-dimensional Euclidean space. This thesis introduces a new integral, the generalized Lebesgue integral, which can be defined on any sigma-finite measure space, and allows the integration of some functions whose absolute values have infinite integrals. The definition retains some of the flavor of the definition of the Lebesgue integral, and introduces two new concepts: ex...

Mathematics, 2022

Broadcasting problems in graph theory play a significant role in solving many complicated physica... more Broadcasting problems in graph theory play a significant role in solving many complicated physical problems. However, in real life there are many vague situations that sometimes cannot be modeled using usual graphs. Consequently, the concept of a fuzzy graph GF:(V,σ,μ) has been introduced to deal with such problems. In this study, we are interested in defining the notion of dominating broadcasts in fuzzy graphs. We also show that, in a connected fuzzy graph containing more than one element in σ*, a dominating broadcast always exists, where σ* is {v∈V|σ(v)>0}. In addition, we investigate the relationship between broadcast domination numbers, radii, and domination numbers in a fuzzy graph as follows; γb(GF)≤min{r(GF),γ(GF)}, where γb(GF) is the broadcast domination number, r(GF) is the radius, and γ(GF) is domination numbers in fuzzy graph GF, with |σ*|>1.

Journal of Differential Equations, 2014

Presented here is a study of a viscoelastic wave equation with supercritical source and damping t... more Presented here is a study of a viscoelastic wave equation with supercritical source and damping terms. We employ the theory of monotone operators and nonlinear semigroups, combined with energy methods to establish the existence of a unique local weak solution. In addition, it is shown that the solution depends continuously on the initial data and is global provided the damping dominates the source in an appropriate sense.

Applicable Analysis, 2010

This article is concerned with the global well-posedness of the critically and degenerately dampe... more This article is concerned with the global well-posedness of the critically and degenerately damped system of nonlinear wave equations in a bounded domain Ω ⊂ ℝ, n = 1, 2, 3, with Dirichlét boundary conditions. The nonlinearities f1(u, v) and f2(u, v) act as a strong source in the system. Under some restriction on the parameters in the system we

Zeitschrift für angewandte Mathematik und Physik

This paper presents a study of the asymptotic behavior of the solutions for the history value pro... more This paper presents a study of the asymptotic behavior of the solutions for the history value problem of a viscoelastic wave equation which features a fading memory term as well as a supercritical source term and a frictional damping term: u tt − k(0)∆u − ∞ 0 k ′ (s)∆u(t − s)ds + |u t | m−1 u t = |u| p−1 u, in Ω × (0, T), u(x, t) = u 0 (x, t), in Ω × (−∞, 0], where Ω is a bounded domain in R 3 with a Dirichlét boundary condition and u 0 represents the history value. A suitable notion of a potential well is introduced for the system, and global existence of solutions is justified provided that the history value u 0 is taken from a subset of the potential well. Also, uniform energy decay rate is obtained which depends on the relaxation kernel −k ′ (s) as well as the growth rate of the damping term. This manuscript complements our previous work [23, 24] where Hadamard well-posedness and the singularity formulation have been studied for the system. It is worth stressing the special features of the model, namely, the source term here has a supercritical growth rate and the memory term accounts to the full past history that goes back to −∞.

Journal of Differential Equations

We investigate a hyperbolic PDE, modeling wave propagation in viscoelastic media, under the influ... more We investigate a hyperbolic PDE, modeling wave propagation in viscoelastic media, under the influence of a linear memory term of Boltzmann type, and a nonlinear damping modeling friction, as well as an energy-amplifying supercritical nonlinear source: u tt − k(0)∆u − ∞ 0 k ′ (s)∆u(t − s)ds + |u t | m−1 u t = |u| p−1 u, in Ω × (0, T), u(x, t) = u 0 (x, t), in Ω × (−∞, 0], where Ω is a bounded domain in R 3 with a Dirichlét boundary condition. The relaxation kernel k is monotone decreasing and k(∞) = 1. We study blow-up of solutions when the source is stronger than dissipations, i.e., p > max{m, k(0)}, under two different scenarios: first, the total energy is negative, and the second, the total energy is positive with sufficiently large quadratic energy. This manuscript is a follow-up work of the paper [30] in which Hadamard well-posedness of this equation has been established in the finite energy space. The model under consideration features a supercritical source and a linear memory that accounts for the full past history as time goes to −∞, which is distinct from other relevant models studied in the literature which usually involve subcritical sources and a finite-time memory.

Nonlinear Analysis Theory Methods Applications, Mar 1, 2010

We focus on the global well-posedness of the system of nonlinear wave equations u tt − ∆u + (d|u|... more We focus on the global well-posedness of the system of nonlinear wave equations u tt − ∆u + (d|u| k + e|v| l)|u t | m−1 u t = f 1 (u, v) v tt − ∆v + (d |v| θ + e |u| ρ)|v t | r−1 v t = f 2 (u, v), in a bounded domain Ω ⊂ R n , n = 1, 2, 3, with Dirichlét boundary conditions. The nonlinearities f 1 (u, v) and f 2 (u, v) act as a strong source in the system. Under some restriction on the parameters in the system we obtain several results on the existence of local solutions, global solutions, and uniqueness. In addition, we prove that weak solutions to the system blow up in finite time whenever the initial energy is negative and the exponent of the source term is more dominant than the exponents of both damping terms.

Thesis (M.Sc.)--Chulalongkorn University, 2001 As a general tool for integration, the Lebesgue in... more Thesis (M.Sc.)--Chulalongkorn University, 2001 As a general tool for integration, the Lebesgue integral is very useful. Its definition is quite general and it has a well-developed theory. However, it does have its limitations. An important limitation is that a function will be Lebesgue integrable only if its absolute value has finite integral, and there exist simple examples of functions that do not satisfy this property, yet intuition suggests they should be integrable. The generalized Riemann integral has helped to solve this problem. Unfortunately, it has a useful theory only for integration over subsets of finite-dimensional Euclidean space. This thesis introduces a new integral, the generalized Lebesgue integral, which can be defined on any sigma-finite measure space, and allows the integration of some functions whose absolute values have infinite integrals. The definition retains some of the flavor of the definition of the Lebesgue integral, and introduces two new concepts: ex...

Mathematics, 2022

Broadcasting problems in graph theory play a significant role in solving many complicated physica... more Broadcasting problems in graph theory play a significant role in solving many complicated physical problems. However, in real life there are many vague situations that sometimes cannot be modeled using usual graphs. Consequently, the concept of a fuzzy graph GF:(V,σ,μ) has been introduced to deal with such problems. In this study, we are interested in defining the notion of dominating broadcasts in fuzzy graphs. We also show that, in a connected fuzzy graph containing more than one element in σ*, a dominating broadcast always exists, where σ* is {v∈V|σ(v)>0}. In addition, we investigate the relationship between broadcast domination numbers, radii, and domination numbers in a fuzzy graph as follows; γb(GF)≤min{r(GF),γ(GF)}, where γb(GF) is the broadcast domination number, r(GF) is the radius, and γ(GF) is domination numbers in fuzzy graph GF, with |σ*|>1.

Journal of Differential Equations, 2014

Presented here is a study of a viscoelastic wave equation with supercritical source and damping t... more Presented here is a study of a viscoelastic wave equation with supercritical source and damping terms. We employ the theory of monotone operators and nonlinear semigroups, combined with energy methods to establish the existence of a unique local weak solution. In addition, it is shown that the solution depends continuously on the initial data and is global provided the damping dominates the source in an appropriate sense.

Applicable Analysis, 2010

This article is concerned with the global well-posedness of the critically and degenerately dampe... more This article is concerned with the global well-posedness of the critically and degenerately damped system of nonlinear wave equations in a bounded domain Ω ⊂ ℝ, n = 1, 2, 3, with Dirichlét boundary conditions. The nonlinearities f1(u, v) and f2(u, v) act as a strong source in the system. Under some restriction on the parameters in the system we

Zeitschrift für angewandte Mathematik und Physik

This paper presents a study of the asymptotic behavior of the solutions for the history value pro... more This paper presents a study of the asymptotic behavior of the solutions for the history value problem of a viscoelastic wave equation which features a fading memory term as well as a supercritical source term and a frictional damping term: u tt − k(0)∆u − ∞ 0 k ′ (s)∆u(t − s)ds + |u t | m−1 u t = |u| p−1 u, in Ω × (0, T), u(x, t) = u 0 (x, t), in Ω × (−∞, 0], where Ω is a bounded domain in R 3 with a Dirichlét boundary condition and u 0 represents the history value. A suitable notion of a potential well is introduced for the system, and global existence of solutions is justified provided that the history value u 0 is taken from a subset of the potential well. Also, uniform energy decay rate is obtained which depends on the relaxation kernel −k ′ (s) as well as the growth rate of the damping term. This manuscript complements our previous work [23, 24] where Hadamard well-posedness and the singularity formulation have been studied for the system. It is worth stressing the special features of the model, namely, the source term here has a supercritical growth rate and the memory term accounts to the full past history that goes back to −∞.

Journal of Differential Equations

We investigate a hyperbolic PDE, modeling wave propagation in viscoelastic media, under the influ... more We investigate a hyperbolic PDE, modeling wave propagation in viscoelastic media, under the influence of a linear memory term of Boltzmann type, and a nonlinear damping modeling friction, as well as an energy-amplifying supercritical nonlinear source: u tt − k(0)∆u − ∞ 0 k ′ (s)∆u(t − s)ds + |u t | m−1 u t = |u| p−1 u, in Ω × (0, T), u(x, t) = u 0 (x, t), in Ω × (−∞, 0], where Ω is a bounded domain in R 3 with a Dirichlét boundary condition. The relaxation kernel k is monotone decreasing and k(∞) = 1. We study blow-up of solutions when the source is stronger than dissipations, i.e., p > max{m, k(0)}, under two different scenarios: first, the total energy is negative, and the second, the total energy is positive with sufficiently large quadratic energy. This manuscript is a follow-up work of the paper [30] in which Hadamard well-posedness of this equation has been established in the finite energy space. The model under consideration features a supercritical source and a linear memory that accounts for the full past history as time goes to −∞, which is distinct from other relevant models studied in the literature which usually involve subcritical sources and a finite-time memory.

Nonlinear Analysis Theory Methods Applications, Mar 1, 2010

We focus on the global well-posedness of the system of nonlinear wave equations u tt − ∆u + (d|u|... more We focus on the global well-posedness of the system of nonlinear wave equations u tt − ∆u + (d|u| k + e|v| l)|u t | m−1 u t = f 1 (u, v) v tt − ∆v + (d |v| θ + e |u| ρ)|v t | r−1 v t = f 2 (u, v), in a bounded domain Ω ⊂ R n , n = 1, 2, 3, with Dirichlét boundary conditions. The nonlinearities f 1 (u, v) and f 2 (u, v) act as a strong source in the system. Under some restriction on the parameters in the system we obtain several results on the existence of local solutions, global solutions, and uniqueness. In addition, we prove that weak solutions to the system blow up in finite time whenever the initial energy is negative and the exponent of the source term is more dominant than the exponents of both damping terms.