Local solutions to semilinear parabolic PDE with a singular nonlinearity

[doobly]

Hi everyone,

I met a specific semilinear second-order PDE given by

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

$u(x,0) = b>0,$

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

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

is a uniformly elliptic differential operator, the coefficients $a_{11}(x,t)...$ 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 $u^{-p},\ p>0$. In addition, the domain $x\in {\bf R}_+^2$ is unbounded. I appreciate that PDE experts would provide some ideas or references on it to me. Thanks again.

 

[jvicens]

Hi there,

I understand your concern about proving the existence and uniqueness of local solutions to this specific PDE. It is definitely a challenging problem due to the singular nonlinearity and unbounded domain. However, there are some techniques and approaches that can be used to tackle this problem.

Firstly, you can try using the method of characteristics to transform the PDE into an ODE and then solve it using standard techniques. This approach has been successful in solving PDEs with singular nonlinearities before. You can also try using the method of separation of variables or the Laplace transform method to simplify the PDE and solve it.

Another approach you can try is using the theory of weak solutions. This theory allows for the existence of solutions to PDEs with singular nonlinearities and unbounded domains. You can refer to the book "Partial Differential Equations" by L.C. Evans for more information on this method.

In addition, there are some numerical methods that can also be used to solve this PDE, such as finite element methods or finite difference methods. These methods can provide approximate solutions to the PDE and can be useful in studying the behavior of the solutions.

I hope these ideas and references will be helpful in your research. Good luck!