Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.1.5
S. Baraket, Safia Mahdaoui, Taieb Ouni
Let (Omega) be a bounded domain in (mathbb{R}^4) with smooth boundary and let (x^{1}, x^{2}, ldots, x^{m}) be (m)-points in (Omega). We are concerned with the problem [Delta^{2} u - H(x,u,D^{k}u) = rho^{4}prod_{i=1}^{n}|x-p_{i}|^{4alpha_{i}}f(x)g(u),] where the principal term is the bi-Laplacian operator, (H(x,u,D^{k}u)) is a functional which grows with respect to (Du) at most like (|Du|^{q}), (1leq qleq 4), (f:Omegato [0,+infty[) is a smooth function satisfying (f(p_{i}) gt 0) for any (i = 1,ldots, n), (alpha_{i}) are positives numbers and (g :mathbb Rto [0,+infty[) satisfy (|g(u)|leq ce^{u}). In this paper, we give sufficient conditions for existence of a family of positive weak solutions ((u_rho)_{rhogt 0}) in (Omega) under Navier boundary conditions (u=Delta u =0) on (partialOmega). The solutions we constructed are singular as the parameters ( ho) tends to 0, when the set of concentration (S={x^{1},ldots,x^{m}}subsetOmega) and the set (Lambda :={p_{1},ldots, p_{n}}subsetOmega) are not necessarily disjoint. The proof is mainly based on nonlinear domain decomposition method.
{"title":"On the blowing up solutions of the 4-d general q-Kuramoto-Sivashinsky equation with exponentially \"dominated\" nonlinearity and singular weight","authors":"S. Baraket, Safia Mahdaoui, Taieb Ouni","doi":"10.7494/opmath.2023.43.1.5","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.1.5","url":null,"abstract":"Let (Omega) be a bounded domain in (mathbb{R}^4) with smooth boundary and let (x^{1}, x^{2}, ldots, x^{m}) be (m)-points in (Omega). We are concerned with the problem [Delta^{2} u - H(x,u,D^{k}u) = rho^{4}prod_{i=1}^{n}|x-p_{i}|^{4alpha_{i}}f(x)g(u),] where the principal term is the bi-Laplacian operator, (H(x,u,D^{k}u)) is a functional which grows with respect to (Du) at most like (|Du|^{q}), (1leq qleq 4), (f:Omegato [0,+infty[) is a smooth function satisfying (f(p_{i}) gt 0) for any (i = 1,ldots, n), (alpha_{i}) are positives numbers and (g :mathbb Rto [0,+infty[) satisfy (|g(u)|leq ce^{u}). In this paper, we give sufficient conditions for existence of a family of positive weak solutions ((u_rho)_{rhogt 0}) in (Omega) under Navier boundary conditions (u=Delta u =0) on (partialOmega). The solutions we constructed are singular as the parameters ( ho) tends to 0, when the set of concentration (S={x^{1},ldots,x^{m}}subsetOmega) and the set (Lambda :={p_{1},ldots, p_{n}}subsetOmega) are not necessarily disjoint. The proof is mainly based on nonlinear domain decomposition method.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342670","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.2.131
Abdulaziz Almaslokh, C. Qian
In the present paper, we study the asymptotic behavior of the following higher order nonlinear difference equation with unimodal terms [x(n+1)= ax(n)+ bx(n)g(x(n)) + cx(n-k)g(x(n-k)), quad n=0, 1, ldots,] where (a), (b) and (c) are constants with (0lt alt 1), (0leq blt 1), (0leq c lt 1) and (a+b+c=1), (gin C[[0, infty), [0, infty)]) is decreasing, and (k) is a positive integer. We obtain some new sufficient conditions for the global attractivity of positive solutions of the equation. Applications to some population models are also given.
本文研究了含单峰项[x(n+1)= ax(n)+ bx(n)g(x(n)) + cx(n-k)g(x(n-k)), quad n=0, 1, ldots,]的高阶非线性差分方程的渐近行为，其中(a)、(b)和(c)为常数，(0lt alt 1)、(0leq blt 1)、(0leq c lt 1)和(a+b+c=1)、(gin C[[0, infty), [0, infty)])为递减，(k)为正整数。得到了该方程正解全局吸引的几个新的充分条件。并给出了在一些人口模型中的应用。
{"title":"Global attractivity of a higher order nonlinear difference equation with unimodal terms","authors":"Abdulaziz Almaslokh, C. Qian","doi":"10.7494/opmath.2023.43.2.131","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.2.131","url":null,"abstract":"In the present paper, we study the asymptotic behavior of the following higher order nonlinear difference equation with unimodal terms [x(n+1)= ax(n)+ bx(n)g(x(n)) + cx(n-k)g(x(n-k)), quad n=0, 1, ldots,] where (a), (b) and (c) are constants with (0lt alt 1), (0leq blt 1), (0leq c lt 1) and (a+b+c=1), (gin C[[0, infty), [0, infty)]) is decreasing, and (k) is a positive integer. We obtain some new sufficient conditions for the global attractivity of positive solutions of the equation. Applications to some population models are also given.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342729","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.2.221
Manabu Naito
We consider the half-linear differential equation [(|x'|^{alpha}mathrm{sgn},x')' + q(t)|x|^{alpha}mathrm{sgn},x = 0, quad t geq t_{0},] under the condition [lim_{ttoinfty}t^{alpha}int_{t}^{infty}q(s)ds = frac{alpha^{alpha}}{(alpha+1)^{alpha+1}}.] It is shown that if certain additional conditions are satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as (ttoinfty).
{"title":"Existence and asymptotic behavior of nonoscillatory solutions of half-linear ordinary differential equations","authors":"Manabu Naito","doi":"10.7494/opmath.2023.43.2.221","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.2.221","url":null,"abstract":"We consider the half-linear differential equation [(|x'|^{alpha}mathrm{sgn},x')' + q(t)|x|^{alpha}mathrm{sgn},x = 0, quad t geq t_{0},] under the condition [lim_{ttoinfty}t^{alpha}int_{t}^{infty}q(s)ds = frac{alpha^{alpha}}{(alpha+1)^{alpha+1}}.] It is shown that if certain additional conditions are satisfied, then the above equation has a pair of nonoscillatory solutions with specific asymptotic behavior as (ttoinfty).","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342990","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.4.547
M. Majewski
In the paper, the Lagrange problem given by a fractional boundary problem with partial derivatives is considered. The main result is the existence of optimal solutions based on the convexity assumption of a certain set. The proof is based on the lower closure theorem and the appropriate implicit measurable function theorem.
{"title":"On the existence of optimal solutions to the Lagrange problem governed by a nonlinear Goursat-Darboux problem of fractional order","authors":"M. Majewski","doi":"10.7494/opmath.2023.43.4.547","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.4.547","url":null,"abstract":"In the paper, the Lagrange problem given by a fractional boundary problem with partial derivatives is considered. The main result is the existence of optimal solutions based on the convexity assumption of a certain set. The proof is based on the lower closure theorem and the appropriate implicit measurable function theorem.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342996","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.5.689
E. C. Lapa
In this paper we consider the existence and asymptotic behavior of solutions of the following nonlinear Kirchhoff type problem [u_{tt}- Mleft(,displaystyle int_{Omega}|nabla u|^{2}, dxright)triangle u - deltatriangle u_{t}= mu|u|^{rho-2}uquad text{in } Omega times ]0,infty[,] where [M(s)=begin{cases}a-bs &text{for } s in [0,frac{a}{b}[, 0, &text{for } s in [frac{a}{b}, +infty[.end{cases}] If the initial energy is appropriately small, we derive the global existence theorem and its exponential decay.
本文考虑以下非线性Kirchhoff型问题[u_{tt}- Mleft(,displaystyle int_{Omega}|nabla u|^{2}, dxright)triangle u - deltatriangle u_{t}= mu|u|^{rho-2}uquad text{in } Omega times ]0,infty[,]解的存在性和渐近性，其中[M(s)=begin{cases}a-bs &text{for } s in [0,frac{a}{b}[, 0, &text{for } s in [frac{a}{b}, +infty[.end{cases}]当初始能量适当小时，我们导出了全局存在性定理及其指数衰减。
{"title":"Global solutions for a nonlinear Kirchhoff type equation with viscosity","authors":"E. C. Lapa","doi":"10.7494/opmath.2023.43.5.689","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.5.689","url":null,"abstract":"In this paper we consider the existence and asymptotic behavior of solutions of the following nonlinear Kirchhoff type problem [u_{tt}- Mleft(,displaystyle int_{Omega}|nabla u|^{2}, dxright)triangle u - deltatriangle u_{t}= mu|u|^{rho-2}uquad text{in } Omega times ]0,infty[,] where [M(s)=begin{cases}a-bs &text{for } s in [0,frac{a}{b}[, 0, &text{for } s in [frac{a}{b}, +infty[.end{cases}] If the initial energy is appropriately small, we derive the global existence theorem and its exponential decay.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71343271","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.3.455
O. Moaaz, M. Anis, A. El-Deeb, Ahmed M. Elshenhab
{"title":"New oscillation constraints for even-order delay differential equations","authors":"O. Moaaz, M. Anis, A. El-Deeb, Ahmed M. Elshenhab","doi":"10.7494/opmath.2023.43.3.455","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.3.455","url":null,"abstract":"","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342528","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.3.429
G. Lau, W. Shiu
Let (G = (V,E)) be a connected simple graph of order (p) and size (q). A graph (G) is called local antimagic (total) if (G) admits a local antimagic (total) labeling. A bijection (g : E to {1,2,ldots,q}) is called a local antimagic labeling of $ if for any two adjacent vertices (u) and (v), we have (g^+(u) ne g^+(v)), where (g^+(u) = sum_{ein E(u)} g(e)), and (E(u)) is the set of edges incident to (u). Similarly, a bijection (f:V(G)cup E(G)to {1,2,ldots,p+q}) is called a local antimagic total labeling of (G) if for any two adjacent vertices (u) and (v), we have (w_f(u)ne w_f(v)), where (w_f(u) = f(u) + sum_{ein E(u)} f(e)). Thus, any local antimagic (total) labeling induces a proper vertex coloring of (G) if vertex (v) is assigned the color (g^+(v)) (respectively, (w_f(u))). The local antimagic (total) chromatic number, denoted (chi_{la}(G)) (respectively (chi_{lat}(G))), is the minimum number of induced colors taken over local antimagic (total) labeling of (G). In this paper, we determined (chi_{lat}(G)) where (G) is the amalgamation ofcomplete graphs. Consequently, we also obtained the local antimagic (total) chromatic number of the disjoint union of complete graphs, and the join of (K_1) and amalgamation of complete graphs under various conditions. An application of local antimagic total chromatic number is also given.
设(G = (V,E))是一个阶为p，大小为q的连通简单图。如果图(G)允许一个局部反魔术(全)标记，则图(G)称为局部反魔术(全)。对于任意两个相邻的点(u)和(v)，我们有(g^+(u) ne g^+(v))，其中(g^+(u) = sum_{E in E(u)} g(E))，而(E(u))是关联到(u)的边的集合。类似地，一个双射(f:V(G)杯E(G)到{1,2，ldots,p+q})被称为(G)的局部反奇异全标记，如果对于任意两个相邻的顶点(u)和(V )，我们有(w_f(u)ne w_f(V))，其中(w_f(u) = f(u) + sum_{E in E(u)} f(E))。因此，如果顶点(v)被赋予颜色(G ^+(v))(分别为(w_f(u)))，则任何局部反奇异(全)标记都会诱导出适当的顶点着色(G)。局部反魔术(总)色数表示为(chi_{la}(G))(分别为(chi_{lat}(G)))，是占据(G)局部反魔术(总)标记的最小诱导色数。在本文中，我们确定了(chi_{lat}(G))，其中(G)是完全图的合并。因此，我们也得到了完全图的不相交并的局部反奇异(全)色数，以及各种条件下完全图的K_1的连接和合并。给出了局部反幻全色数的一个应用。
{"title":"On local antimagic total labeling of complete graphs amalgamation","authors":"G. Lau, W. Shiu","doi":"10.7494/opmath.2023.43.3.429","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.3.429","url":null,"abstract":"Let (G = (V,E)) be a connected simple graph of order (p) and size (q). A graph (G) is called local antimagic (total) if (G) admits a local antimagic (total) labeling. A bijection (g : E to {1,2,ldots,q}) is called a local antimagic labeling of $ if for any two adjacent vertices (u) and (v), we have (g^+(u) ne g^+(v)), where (g^+(u) = sum_{ein E(u)} g(e)), and (E(u)) is the set of edges incident to (u). Similarly, a bijection (f:V(G)cup E(G)to {1,2,ldots,p+q}) is called a local antimagic total labeling of (G) if for any two adjacent vertices (u) and (v), we have (w_f(u)ne w_f(v)), where (w_f(u) = f(u) + sum_{ein E(u)} f(e)). Thus, any local antimagic (total) labeling induces a proper vertex coloring of (G) if vertex (v) is assigned the color (g^+(v)) (respectively, (w_f(u))). The local antimagic (total) chromatic number, denoted (chi_{la}(G)) (respectively (chi_{lat}(G))), is the minimum number of induced colors taken over local antimagic (total) labeling of (G). In this paper, we determined (chi_{lat}(G)) where (G) is the amalgamation ofcomplete graphs. Consequently, we also obtained the local antimagic (total) chromatic number of the disjoint union of complete graphs, and the join of (K_1) and amalgamation of complete graphs under various conditions. An application of local antimagic total chromatic number is also given.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342910","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.6.803
J. Hejduk, Renata Wiertelak
This paper deals with essential generalization of (mathcal{I})-density points and (mathcal{I})-density topology. In particular, there is an example showing that this generalization of (mathcal{I})-density point yields the stronger concept of density point than the notion of (mathcal{I}(mathcal{J}))-density. Some properties of topologies generated by operators related to this essential generalization of density points are provided.
{"title":"On the concept of generalization of I-density points","authors":"J. Hejduk, Renata Wiertelak","doi":"10.7494/opmath.2023.43.6.803","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.6.803","url":null,"abstract":"This paper deals with essential generalization of (mathcal{I})-density points and (mathcal{I})-density topology. In particular, there is an example showing that this generalization of (mathcal{I})-density point yields the stronger concept of density point than the notion of (mathcal{I}(mathcal{J}))-density. Some properties of topologies generated by operators related to this essential generalization of density points are provided.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71343323","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.6.865
Michal Sta�, M�ria Timkov�
. The crossing number cr( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. In the paper, we extend known results concerning crossing numbers of join products of four small graphs with paths and cycles. The crossing numbers of the join products G ∗ + P n and G ∗ + C n for the disconnected graph G ∗ consisting of the complete tripartite graph K 1 , 1 , 2 and one isolated vertex are given, where P n and C n are the path and the cycle on n vertices, respectively. In the paper also the crossing numbers of H ∗ + P n and H ∗ + C n are determined, where H ∗ is isomorphic to the complete tripartite graph K 1 , 1 , 3 . Finally, by adding new edges to the graphs G ∗ and H ∗ , we are able to obtain crossing numbers of join products of two other graphs G 1 and H 1 with paths and cycles.
． 图G的交叉数cr(G)是平面上所有图G的最小边交叉数。本文推广了关于四个带路径和环的小图的连接积相交数的已知结果。给出了由完全三部图k1,1,2和一个孤立顶点组成的不连通图G∗+ pn和G∗+ cn的连接积G∗+ pn和G∗+ cn的交叉数，其中pn和cn分别是n个顶点上的路径和循环。文中还确定了H∗+ P n和H∗+ C n的交点数，其中H∗同构于完全三部图k1,1,3。最后，通过在图G∗和H∗上添加新的边，我们能够得到另外两个图g1和h1具有路径和环的连接积的交叉数。
{"title":"The crossing numbers of join products of four graphs of order five with paths and cycles","authors":"Michal Sta�, M�ria Timkov�","doi":"10.7494/opmath.2023.43.6.865","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.6.865","url":null,"abstract":". The crossing number cr( G ) of a graph G is the minimum number of edge crossings over all drawings of G in the plane. In the paper, we extend known results concerning crossing numbers of join products of four small graphs with paths and cycles. The crossing numbers of the join products G ∗ + P n and G ∗ + C n for the disconnected graph G ∗ consisting of the complete tripartite graph K 1 , 1 , 2 and one isolated vertex are given, where P n and C n are the path and the cycle on n vertices, respectively. In the paper also the crossing numbers of H ∗ + P n and H ∗ + C n are determined, where H ∗ is isomorphic to the complete tripartite graph K 1 , 1 , 3 . Finally, by adding new edges to the graphs G ∗ and H ∗ , we are able to obtain crossing numbers of join products of two other graphs G 1 and H 1 with paths and cycles.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71343343","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2023-01-01DOI: 10.7494/opmath.2023.43.3.409
M. Dilmi, Sadok Otmani
In this paper we propose a new mathematical model describing the deformations of an isotropic nonlinear elastic body with variable exponent in dynamic regime. We assume that the stress tensor (sigma^{p(cdot)}) has the form [sigma^{p(cdot)}(u)=(2mu +|d(u)|^{p(cdot)-2})d(u)+lambda Tr(d(u)) I_{3},] where (u) is the displacement field, (mu), (lambda) are the given coefficients (d(cdot)) and (I_{3}) are the deformation tensor and the unit tensor, respectively. By using the Faedo-Galerkin techniques and a compactness result we prove the existence of the weak solutions, then we study the asymptotic behaviour stability of the solutions.
{"title":"Existence and asymptotic stability for generalized elasticity equation with variable exponent","authors":"M. Dilmi, Sadok Otmani","doi":"10.7494/opmath.2023.43.3.409","DOIUrl":"https://doi.org/10.7494/opmath.2023.43.3.409","url":null,"abstract":"In this paper we propose a new mathematical model describing the deformations of an isotropic nonlinear elastic body with variable exponent in dynamic regime. We assume that the stress tensor (sigma^{p(cdot)}) has the form [sigma^{p(cdot)}(u)=(2mu +|d(u)|^{p(cdot)-2})d(u)+lambda Tr(d(u)) I_{3},] where (u) is the displacement field, (mu), (lambda) are the given coefficients (d(cdot)) and (I_{3}) are the deformation tensor and the unit tensor, respectively. By using the Faedo-Galerkin techniques and a compactness result we prove the existence of the weak solutions, then we study the asymptotic behaviour stability of the solutions.","PeriodicalId":45563,"journal":{"name":"Opuscula Mathematica","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"71342862","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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