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Local solutions to semilinear parabolic PDE with a singular nonlinearity

  1. Jan 31, 2013 #1
    Hi everyone,

    I met a specific semilinear second-order PDE given by

    [itex]\frac{\partial u(x,t)}{\partial t} = {\bf A}(x,t) u(x,t) + U(x,t) u(x,t)^{-p} + A(x,t),....(1) [/itex]

    [itex] u(x,0) = b>0, [/itex]

    where [itex] p>0,\ \ (x,t) = (x_1,x_2,t)\in{\bf R}_+^2\times [0,T] [/itex], and

    [itex] {\bf A}(x,t)u = a_{11}(x,t)\frac{\partial^2 u}{\partial x_1^2} + a_{22}(x,t)\frac{\partial^2 u}{\partial x_2^2} + a_1(x,t)\frac{\partial u}{\partial x_1}
    + a_2(x,t)\frac{\partial u}{\partial x_2} + a_0(x,t) u [/itex]

    is a uniformly elliptic differential operator, the coefficients [itex] a_{11}(x,t)... [/itex] are smooth enough w.r.t. (x,t)

    Both U(x,t) and A(x,t) are positive for all (x,t) and which are also smooth w.r.t. (x,t).

    Now I want to prove the existence and uniqueness of local solutions to (1). However, in (1), there is a singular nonlinearity [itex] u^{-p},\ p>0[/itex]. In addition, the domain [itex] x\in {\bf R}_+^2[/itex] is unbounded. I appreciate that PDE experts would provide some ideas or references on it to me. Thanks again.
  2. jcsd
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