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This thesis will present new results involving Hardy and Hardy-Sobolev-Maz'ya inequalities for fractional integrals. There are two key ingredients to many of these results. The first is the conformal transformation between the upper halfspace and the unit ball. The second is the pseudosymmetric halfspace rearrangement, which is a type of rearrangment on the upper halfspace based on Carlen and Loss' concept of competing symmetries along with certain geometric considerations from the conformal transformation. After reducing to one dimension, we can use the conformal transformation to prove a sharp Hardy inequality for general domains, as well as an improved fractional Hardy inequality over convex domains. Most importantly, the sharp constant is the same as that for the halfspace. Two new Hardy-Sobolev-Maz'ya inequalities will also be established. The first will be a weighted inequality that has a strong relationship with the pseudosymmetric halfspace rearrangement. Then, the psuedosymmetric halfspace rearrangement will play a key part in proving the existence of the standard Hardy-Sobolev-Maz'ya inequality on the halfspace, as well as some results involving the existence of minimizers for that inequality.

By introducing a concept of dynamic process operating under multi-time scales in sciences and engineering, a mathematical model is formulated and it leads to a system of multi-time scale stochastic differential equations. The classical Picard-Lindel\"{o}f successive approximations scheme is expended to the model validation problem, namely, existence and uniqueness of solution process. Naturally, this generates to a problem of finding closed form solutions of both linear and nonlinear multi-time scale stochastic differential equations. To illustrate the scope of ideas and presented results, multi-time scale stochastic models for ecological and epidemiological processes in population dynamic are exhibited. Without loss in generality, the modeling and analysis of three time-scale fractional stochastic differential equations is followed by the development of the numerical algorithm for multi-time scale dynamic equations. The development of numerical algorithm is based on the idea if numerical integration in the context of the notion of multi-time scale integration. The multi-time scale approach is applied to explore the study of higher order stochastic differential equations (HOSDE) is presented. This study utilizes the variation of constant parameter technique to develop a method for finding closed form solution processes of classes of HOSDE. Then then probability distribution of the solution processes in the context of the second order equations is investigated.

This thesis presents results in two apparently disparate mathematical fields which can both be examined -- and even united -- by means of pure analysis. Fractional calculus is the study of differentiation and integration to non-integer orders. Dating back to Leibniz, this idea was considered by many great mathematical figures, and in recent decades it has been used to model many real-world systems and processes, but a full development of the mathematical theory remains incomplete. Many techniques for partial differential equations (PDEs) can be extended to fractional PDEs too. Three chapters below cover my results in this area: establishing the elliptic regularity theorem, Malgrange-Ehrenpreis theorem, and unified transform method for fractional PDEs. Each one is analogous to a known result for classical PDEs, but the proof in the general fractional scenario requires new ideas and modifications. Fractional derivatives and integrals are not uniquely defined: there are many different formulae, each of which has its own advantages and disadvantages. The most commonly used is the classical Riemann-Liouville model, but others may be preferred in different situations, and now new fractional models are being proposed and developed each year. This creates many opportunities for new research, since each time a model is proposed, its mathematical fundamentals need to be examined and developed. Two chapters below investigate some of these new models. My results on the Atangana-Baleanu model proposed in 2016 have already had a noticeable impact on research in this area. Furthermore, this model and the results concerning it can be extended to more general fractional models which also have certain desirable properties of their own. Fractional calculus and zeta functions have rarely been united in research, but one chapter below covers a new formula expressing the Lerch zeta function as a fractional derivative of an elementary function. This result could have many ramifications in both fields, which are yet to be explored fully. Zeta functions are very important in analytic number theory: the Riemann zeta function relates to the distribution of the primes, and this field contains some of the most persistent open problems in mathematics. Since 2012, novel asymptotic techniques have been applied to derive new results on the growth of the Riemann zeta function. One chapter below modifies some of these techniques to prove asymptotics to all orders for the Hurwitz zeta function. Many new ideas are required, but the end result is more elegant than the original one for Riemann zeta, because some of the new methodologies enable different parts of the argument to be presented in a more unified way. Several related problems involve asymptotics arbitrarily near a stationary point. Ideally it should be possible to find uniform asymptotics which provide a smooth transition between the integration by parts and stationary phase methods. One chapter below solves this problem for a particular integral which arises in the analysis of zeta functions.

Dans cette thèse, nous nous intéressons à l'étude de trois équations aux dérivées partielles et d'évolution non-locales en espace et en temps. Les solutions de ces trois solutions peuvent exploser en temps fini. Dans une première partie de cette thèse, nous considérons l'équation de la chaleur nonlinéaire avec une puissance fractionnaire du laplacien, et obtenons notamment que, dans le cas d'exposant sur-critique, le comportement asymptotique de la solution lorsque $t\rightarrow+\infty$ est déterminé par le terme de diffusion anormale. D'autre part, dans le cas d'exposant sous-critique, l'effet du terme non-linéaire domine. Dans une deuxième partie, nous étudions une équation parabolique avec le laplacien fractionnaire et un terme non-linéaire et non-local en temps. On montre que la solution est globale dans le cas sur-critique pour toute donnée initiale ayant une mesure assez petite, tandis que dans le cas sous-critique, on montre que la solution explose en temps fini $T_{\max}>0$ pour toute condition initiale positive et non-triviale. Dans ce dernier cas, on cherche le comportement de la norme $L^1$ de la solution en précisant le taux d'explosion lorsque $t$ s'approche du temps d'explosion $T_{\max}.$ Nous cherchons encore les conditions nécessaires à l'existence locale et globale de la solution. Une toisième partie est consacré à une généralisation de la deuxième partie au cas de systèmes $2\times 2$ avec le laplacien ordinaire. On étudie l'existence locale de la solution ainsi qu'un résultat sur l'explosion de la solution avec les mêmes propriétés étudiées dans le troisième chapitre. Dans la dernière partie, nous étudions une équation hyperbolique dans $\mathbb{R}^N,$ pour tout $N\geq2,$ avec un terme non-linéaire non-local en temps. Nous obtenons un résultat d'existence locale de la solution sous des conditions restrictives sur les données initiales, la dimension de l'espace et les exposants du terme non-linéaire. De plus on obtient, sous certaines conditions sur les exposants, que la solution explose en temps fini, pour toute condition initiale ayant de moyenne strictement positive.

Dans cette thèse, nous nous intéressons à l'étude de quatre équations d'évolution non-locales. Les solutions de ces quatre équations peuvent exploser en temps fini. Dans la théorie des équations d'évolution non-linéaires, une solution est qualifiée de globale si elle est définie pour tout temps positif. Au contraire, si une solution existe seulement sur un intervalle de temps [0; T) borné, elle est dite locale. Dans ce dernier cas et quand le temps maximal d'existence est relié à une alternative d'explosion, on dit aussi que la solution explose en temps fini. Dans un premier travail, nous considérons l'équation de Schrödinger non-linéaire avec une puissance fractionnaire du laplacien, et nous obtenons l'explosion de la solution en temps fini Tmax > 0 pour toute condition initiale positive et non-triviale dans le cas d'exposant sous-critique. Ensuite, nous étudions une équation des ondes amorties avec un potentiel d'espace-temps et un terme non-linéaire et non-local en temps. Nous obtenons un résultat d'existence locale d'une solution dans l'espace d'énergie sous des conditions restrictives sur les données initiales, la dimension de l'espace et la croissance du terme non-linéaire. De plus, nous obtenons l'explosion de la solution en temps fini pour toute condition initiale de moyenne strictement positive. De plus, nous étudions un problème de Cauchy pour l'équation d'évolution avec un p- Laplacien avec une non linéarité non-locale en temps. Dans ce cadre, nous nous intéressons à l'étude de l'existence locale d'une solution de cette équation ainsi qu'un résultat de non-existence de solution globale. Finalement, nous étudions l'intervalle maximal d'existence des solutions de l'équation des milieux poreux avec un terme non-linéaire non-local en temps. / In this thesis, we study four non-local evolution equations. The solutions of these four equations can blow up in finite time. In the theory of nonlinear evolution equations, a solution is qualified as global if it isdefined for any time. Otherwise, if a solution exists only on a bounded interval [0; T), it is called local solution. In this case and when the maximum time of existence is related to a blow up alternative, we say that the solution blows up in finite time. First, we consider the nonlinear Schröodinger equation with a fractional power of the Laplacien operator, and we get a blow up result in finite time Tmax > 0 for any non-trivial non-negative initial condition in the case of sub-critical exponent. Next, we study a damped wave equation with a space-time potential and a non-local in time non-linear term. We obtain a result of local existence of a solution in the energy space under some restrictions on the initial data, the dimension of the space and the growth of nonlinear term. Additionally, we get a blow up result of the solution in finite time for any initial condition positive on average. In addition, we study a Cauchy problem for the evolution p-Laplacien equation with nonlinear memory. We study the local existence of a solution of this equation as well as a result of non-existence of global solution. Finally, we study the maximum interval of existence of solutions of the porous medium equation with a nonlinear non-local in time term.