Mathematical Sciences: On the Existence and Qualitative Properties of Solutions of Nonlinear Partical Differential Equations

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The focus of this work will be the study of monotonicity and symmetry properties of solutions of partial differential equations. Equations of interest include those arising in modeling vibrations and in nonlinear Schrodinger equations. They fall into the classes known as fully nonlinear elliptic and parabolic equations. One is interested in understanding solutions where the minimal period is prescribed or the nature of the singularity where a solution ceases to exist (blow up). Progress in the first case has been made; the major obstacle to be overcome rests with the complexity of the infinite dimensional kernel of the wave operator. Development in the study of blow up of solutions of the Schrodinger equation have advanced considerably in the past decade. It has been shown that when the spatial domain is star-shaped, solutions do blow up in finite time. Work will continue in this vein to understand better the nature of the singularity and to obtain sharper estimates regarding the time at which the blow up can be expected. Currently, many related estimates are very crude. The onset of blow up is not yet known to have physical significance because the minimal possible time may be exponentially large. Other work will investigate the existence, uniqueness and spatial decay properties of solutions of traveling waves arising from dynamics and combustion problems.