# Local solutions to semilinear parabolic PDE with a singular nonlinearity

• doobly
In summary, the conversation discussed a specific semilinear second-order PDE with a singular nonlinearity and unbounded domain. The challenge is to prove the existence and uniqueness of local solutions to this PDE. Possible approaches include using the method of characteristics, separation of variables, Laplace transform, weak solutions theory, and numerical methods. Reference to the book "Partial Differential Equations" by L.C. Evans was also provided as a helpful resource.
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.

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!

## 1. What is a semilinear parabolic PDE with a singular nonlinearity?

A semilinear parabolic PDE (partial differential equation) is a mathematical model that describes the evolution of a physical system over time, taking into account both the spatial and temporal variations of the system. The term "parabolic" refers to the shape of the equation's graph, which is often a parabola. A singular nonlinearity in this context means that the equation has a term that becomes infinite at a certain point, leading to a singularity in the solution.

## 2. What are local solutions in the context of this type of PDE?

A local solution to a PDE is a solution that is valid only in a specific region of space and time. In other words, it is a solution that is valid within a certain domain, but may not be valid outside of that domain. For a semilinear parabolic PDE with a singular nonlinearity, the local solution is valid only in a certain region where the singularity does not occur.

## 3. How do you find local solutions to semilinear parabolic PDEs with a singular nonlinearity?

The process of finding local solutions to these types of PDEs involves using various mathematical techniques, such as separation of variables, change of variables, and Fourier series. These techniques help to transform the original PDE into a simpler form that can be solved using known methods, such as integration or solving an ordinary differential equation.

## 4. What are some applications of local solutions to semilinear parabolic PDEs with a singular nonlinearity?

These types of PDEs have various applications in different fields of science and engineering. For example, they are commonly used to model heat transfer, diffusion, and chemical reactions. In these applications, the local solution helps to understand how the system behaves in a specific region, which can be useful for making predictions and designing experiments.

## 5. What are the challenges in studying local solutions to semilinear parabolic PDEs with a singular nonlinearity?

One of the main challenges in studying these types of PDEs is the presence of singularities, which can make the equations difficult to solve. Another challenge is that the solutions may not be unique, meaning that there can be multiple solutions that satisfy the PDE. Additionally, the behavior of the solutions near the singularity can be complex and require advanced mathematical techniques to understand.

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