We set up a framework to study one-dimensional heat equations defined by fractal Laplacians associated with self-similar measures with overlaps. We show that for a class of such self-similar measures, a heat equation can be discretized and the finite element method can be applied to yield a system of linear differential equations. We show that the numerical solutions converge to the actual solution and obtain the rate of convergence. We also study some properties of the solutions of the heat equation.

Light bullets are spatially localized ultra-short optical pulses in more than one space dimensions. They contain only a few electromagnetic oscillations under their envelopes and propagate long distances without essentially changing shapes. Light bullets of femtosecond durations have been observed in recent numerical simulation of the full Maxwell systems. The sineGordon (SG) equation comes as an asymptotic reduction of the two level dissipationless MaxwellBloch system. We derive a new and complete nonlinear Schrdinger (NLS) equation in two space dimensions for the SG pulse envelopes so that it is globally well-posed and has all the relevant higher order terms to regularize the collapse of the standard critical NLS (CNLS). We perform a modulation analysis and found that SG pulse envelopes undergo focusingdefocusing cycles. Numerical results are in qualitative agreement with asymptotics and

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In this paper, we discuss the Weyl problem in warped product spaces. We apply the method of continuity and prove the openness of the Weyl problem. A counterexample is constructed to show that the isometric embedding of the sphere with canonical metric is not unique up to an isometry if the ambient warped product space is not a space form. Then, we study the rigidity of the standard sphere if we fix its geometric center in the ambient space. Finally, we discuss a Shi-Tam type of inequality for the Schwarzschild manifold as an application of our findings.