New development of the qualitative theory of nonlinear parabolic and elliptic equations

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We carried out the investigation about the structure of solutions of nonlinear parabolic and elliptic equations. Our main results are as follows : Next, we studied the existence and uniqueness of solutions with moving singularities for a nonlinear parabolic partial differential equation. We also showed that there exists a solution with a moving singularity that changes its type suddenly., and made clear the asymptotic behavior of singular solutions that converges to a singular steady state. We also studied a chemotaxis system, and made clear the structure of self-similar solutions that blows up by concentrating to a point in finite time.For a reaction-diffusion system, which is called a Gierer-Meinhardt system, we studied the mathematical structure of pattern formation, and also made clear the behavior of time-dependent solutions.