Analysis and Applied Mathematics Seminar
Fred Weissler
Universite de Paris Nord
Anti-symmetric solutions of the nonlinear heat equation on R^n : local existence and finite time blowup.
Abstract: Abstract : We consider a nonlinear heat equation on $R^n$ with a homogeneous, superlinear nonlinearity.
We study solutions which are anti-symmetric with respect to the spatial variables.
It is shown that very singular initial values, e.g. derivatives of the Dirac delta function,
give rise to local (regular) solutions. These solutions exist when the homogeneity of the nonlinearity
is below a value which is consistent with the scaling properties of the equation.
These results enable us to obtain new finite time blowup results for certain classes
of regular anti-symmetric initial values of the form $u_0 = \lambda f$.
Counterintuitively, these blow-up results hold for $\lambda > 0$ sufficiently small.
Blowup results of this type, where the initial value has a small coefficient, were
first found by Dickstein [1]. The work to be presented is joint with S. Tayachi [2, 3].
References
[1] F. Dickstein, Blowup stability of solutions of the nonlinear heat equation with a large life span,
J. Di erential Equations 223 (2006), 303{328.
[2] S. Tayachi and F. B. Weissler, The nonlinear heat equation with high order mixed derivatives
of the Dirac Delta as initial values, Trans. Amer. Math. Soc. 366 (2014), 505-530.
[3] S. Tayachi and F. B. Weissler, The nonlinear heat equation involving highly singular initial
values and new blowup and life span results, Journal of Elliptic and Parabolic Equations, 4
(2018), 141-17
Monday October 21, 2019 at 4:00 PM in 636 SEO