Una contribución al análisis de las ecuaciones en derivadas parciales estocásticas funcionales con derivadas fraccionarias en tiempo y aplicaciones

On the one hand, the classical heat equation∂tu= ∆udescribes heatpropagation in a homogeneous medium, while the time fractional diffusionequation∂αtu= ∆uwith 0< α <1 has been widely used to model anoma-lous diffusion exhibiting subdiffusive behavior. On the other side, when weconsider a physical system in the real world, we have to consider some in-fluences of internal, external, or environmental noises. Besides, the wholebackground of physical system may be difficult to describe deterministical-ly. Therefore, in this thesis, we will construct three models to show theapplications of the time fractional stochastic functional partial differentialequations.In Chapter 2, we study a stochastic lattice system with Caputo fractionalsubstantial time derivative, the asymptotic behavior of this kind of problemis investigated. In particular, the existence of a global forward attractingset in the weak mean-square topology is established. A general theorem onthe existence of solutions for a fractional SDE in a Hilbert space under theassumption that the nonlinear term is weakly continuous in a given sense isestablished and applied to the lattice system. The existence and uniquenessof solutions for a more general fractional SDEs is also obtained under aLipschitz condition.In Chapter 3, the local and global existence and uniqueness of mild solu-tions to a kind of stochastic time fractional impulsive differential equationsare studied by means of a fixed point theorem, and with the help of theproperty ofα-order fractional solution operatorTα(t) and the resolvent op-eratorSα(t). Moreover, the exponential decay to zero of the mild solutionsto this model is also proved. However, the lack of compactness of theα-order resolvent operatorSα(t) does not allow us to establish the existenceand structure of attracting sets, which is a key concept for understandingthe dynamical properties.Therefore, the second model of Chapter 3 is concerned with the well-posedness and dynamics of delay impulsive fractional stochastic evolutionequations with time fractional differential operatorα∈(0,1). After estab-lishing the well-posedness of the problem, and a result ensuring the existenceand uniqueness of mild solutions globally defined in future, the existence ofa minimal global attracting set is investigated in the mean-square topology,under general assumptions not ensuing the uniqueness of solutions. Further-more, in the case of uniqueness, it is possible to provide more informationabout the geometrical structure of such global attracting set. In particular,it is proved that the minimal compact globally attracting set for the solution-1 s of the problem becomes a singleton. It is remarkable that the attractionproperty is proved in the usual forward sense, unlike the pullback conceptused in the context of random dynamical systems, but the main point is thatthe model under study has not been proved to generate a random dynamicalsystem.Chapter 4 is devoted to the well-posedness of stochastic time fractional2D-Stokes equations of orderα∈(0,1) containing finite or infinite delay withmultiplicative noise is established, respectively, in the spacesC([−h,0];L2(Ω);L2σ)) andC((−∞,0];L2(Ω;L2σ)). The existence and uniqueness of mild so-lution to such kind of equations are proved by using a fixed-point argument.Also the continuity with respect to initial data is shown. Finally, we con-clude with several comments on future research concerning the challengingmodel: time fractional stochastic delay 2D-Navier-Stokes equations withmultiplicative noise

Dynamics of stochastic nonlocal partial differential equations

This paper is concerned with the asymptotic behavior of solutions to nonlocal stochastic partial differential equations with multiplicative and additive noise driven by a standard Brownian motion, respectively. First of all, the stochastic nonlocal differential equations are transformed into their associated conjugated random differential equations, we then construct the dynamical systems to the original problems via the properties of conjugation. Next, in the case of multiplicative noise, we establish the existence of the random attractor when it absorbs every bounded deterministic set. Particularly, it is shown the pullback random attractor, which is also forward attracting, becomes a singleton when the external forcing term vanishes at zero. Eventually, in the case of additive noise, two approaches are applied to prove the existence of pullback random attractors with the help of energy estimations. Actually, these two attractors turn out to be the same one

Long time behavior of fractional impulsive stochastic differential equations with infinite delay

This paper is first devoted to the local and global existence of mild solutions for a class of fractional impulsive stochastic differential equations with infinite delay driven by both K-valued Q-cylindrical Brownian motion and fractional Brownian motion with Hurst parameter H ∈ (1/2, 1). A general framework which provides an effective way to prove the continuous dependence of mild solutions on initial value is established under some appropriate assumptions. Furthermore, it is also proved the exponential decay to zero of solutions to fractional stochastic impulsive differential equations with infinite delay.European Commission (EC). Fondo Europeo de Desarrollo Regional (FEDER)Ministerio de Economía y Competitividad (MINECO). EspañaConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía

Long Time Behavior of Stochastic Nonlocal Partial Differential Equations and Wong--Zakai Approximations

This paper is devoted to investigating the well-posedness and asymptotic behavior of a class of stochastic 5 nonlocal partial differential equations driven by nonlinear noise. First, the existence of a weak martingale solution is estab 6 lished by using the Faedo-Galerkin approximation and an idea analogous to Da Prato and Zabczyk [12]. Second, we show 7 the uniqueness and continuous dependence on initial values of solutions to the above stochastic nonlocal problem when there 8 exist some variational solutions. Third, the asymptotic local stability of steady-state solutions is analyzed either when the 9 steady-state solutions of the deterministic problem is also solution of the stochastic one, or when this does not happen. Next, 10 to study the global asymptotic behavior, namely, the existence of attracting sets of solutions, we consider an approximation 11 of the noise given by Wong-Zakai’s technique using the so called colored noise. For this model, we can use the power of 12 the theory of random dynamical systems and prove the existence of random attractors. Eventually, particularizing in the 13 cases of additive and multiplicative noise, it is proved that the Wong-Zakai approximation models possess random attractors 14 which converge upper-semicontinuously to the respective random attractors of the stochastic equations driven by standard 15 Brownian motions. This fact justifies the use of this colored noise technique to approximate the asymptotic behavior of the 16 models with general nonlinear noises, although the convergence of attractors and solutions is still an open problem

Well-posedness and dynamics of impulsive fractional stochastic evolution equations with unbounded delay

This paper is concerned with the well-posedness and dynamics of delay impulsive fractional stochastic evolution equations with time fractional differential operator α ∈ (0, 1). After establishing the well-posedness of the problem, and a result ensuring the existence and uniqueness of mild solutions globally defined in future, the existence of a minimal global attracting set is investigated in the mean-square topology, under general assumptions not ensuing the uniqueness of solutions. Furthermore, in the case of uniqueness, it is possible to provide more information about the geometrical structure of such global attracting set. In particular, it is proved that the minimal compact globally attracting set for the solutions of the problem becomes a singleton. It is remarkable that the attraction property is proved in the usual forward sense, unlike the pullback concept used in the context of random dynamical systems, but the main point is that the model under study has not been proved to generate a random dynamical system.National Natural Science Foundation of ChinaFondo Europeo de Desarrollo Regional (FEDER)Ministerio de Economía y CompetitividadConsejería de Innovación, Ciencia y Empresa (Junta de Andalucía

Mild Solutions to Time Fractional Stochastic 2D-Stokes Equations with Bounded and Unbounded Delay

In this paper, the well-posedness of stochastic time fractional 2D-Stokes equations of order α ∈ (0, 1) containig finite or infinite delay with multiplicative noise is established, respectively, in the spaces C([−h, 0]; L2(Ω; L2 σ )) and C((−∞, 0]; L2(Ω; L2 σ )). The existence and uniqueness of mild solution to such kind of equations are proved by using a fixed-point argument. Also the continuity with respect to initial data is shown. Finally, we conclude with several comments on future research concerning the challenging model: time fractional stochastic delay 2D-Navier–Stokes equations with multiplicative noise. Hence, this paper can be regarded as a first step to study this challenging topic

Asymptotic behavior of a semilinear problem in heat conduction with long time memory and non-local diffusion

In this paper, the asymptotic behavior of a semilinear heat equation with long time memory and non-local diffusion is analyzed in the usual set-up for dynamical systems generated by differential equations with delay terms. This approach is different from the previous published literature on the long time behavior of heat equations with memory which is carried out by the Dafermos transformation. As a consequence, the obtained results provide complete information about the attracting sets for the original problem, instead of the transformed one. In particular, the proved results also generalize and complete previous literature in the local case

Lineage tracing for multiple lung cancer by spatiotemporal heterogeneity using a multi-omics analysis method integrating genomic, transcriptomic, and immune-related features

IntroductionThe distinction between multiple primary lung cancer (MPLC) and intrapulmonary metastasis (IPM) holds clinical significance in staging, therapeutic intervention, and prognosis assessment for multiple lung cancer. Lineage tracing by clinicopathologic features alone remains a clinical challenge; thus, we aimed to develop a multi-omics analysis method delineating spatiotemporal heterogeneity based on tumor genomic profiling.MethodsBetween 2012 and 2022, 11 specimens were collected from two patients diagnosed with multiple lung cancer (LU1 and LU2) with synchronous/metachronous tumors. A novel multi-omics analysis method based on whole-exome sequencing, transcriptome sequencing (RNA-Seq), and tumor neoantigen prediction was developed to define the lineage. Traditional clinicopathologic reviews and an imaging-based algorithm were performed to verify the results.ResultsSeven tissue biopsies were collected from LU1. The multi-omics analysis method demonstrated that three synchronous tumors observed in 2018 (LU1B/C/D) had strong molecular heterogeneity, various RNA expression and immune microenvironment characteristics, and unique neoantigens. These results suggested that LU1B, LU1C, and LU1D were MPLC, consistent with traditional lineage tracing approaches. The high mutational landscape similarity score (75.1%), similar RNA expression features, and considerable shared neoantigens (n = 241) revealed the IPM relationship between LU1F and LU1G which were two samples detected simultaneously in 2021. Although the multi-omics analysis method aligned with the imaging-based algorithm, pathology and clinicopathologic approaches suggested MPLC owing to different histological types of LU1F/G. Moreover, controversial lineage or misclassification of LU2’s synchronous/metachronous samples (LU2B/D and LU2C/E) traced by traditional approaches might be corrected by the multi-omics analysis method. Spatiotemporal heterogeneity profiled by the multi-omics analysis method suggested that LU2D possibly had the same lineage as LU2B (similarity score, 12.9%; shared neoantigens, n = 71); gefitinib treatment and EGFR, TP53, and RB1 mutations suggested the possibility that LU2E might result from histology transformation of LU2C despite the lack of LU2C biopsy and its histology. By contrast, histological interpretation was indeterminate for LU2D, and LU2E was defined as a primary or progression lesion of LU2C by histological, clinicopathologic, or imaging-based approaches.ConclusionThis novel multi-omics analysis method improves the accuracy of lineage tracing by tracking the spatiotemporal heterogeneity of serial samples. Further validation is required for its clinical application in accurate diagnosis, disease management, and improving prognosis

Draft genome sequence of the Tibetan antelope

The Tibetan antelope (Pantholops hodgsonii) is endemic to the extremely inhospitable high-altitude environment of the Qinghai-Tibetan Plateau, a region that has a low partial pressure of oxygen and high ultraviolet radiation. Here we generate a draft genome of this artiodactyl and use it to detect the potential genetic bases of highland adaptation. Compared with other plain-dwelling mammals, the genome of the Tibetan antelope shows signals of adaptive evolution and gene-family expansion in genes associated with energy metabolism and oxygen transmission. Both the highland American pika, and the Tibetan antelope have signals of positive selection for genes involved in DNA repair and the production of ATPase. Genes associated with hypoxia seem to have experienced convergent evolution. Thus, our study suggests that common genetic mechanisms might have been utilized to enable high-altitude adaptation