Dear mathoverflowers, I have a question concerning the strong convergence in $L^p([0,T],X)$. Let $X_1,X$ be two Banach spaces such that $X_1\subset X$ with compact embedding. Let …

Hi there. Does there exist a maximum principle for the non-uniformly parabolic operator $P = \partial_t - e^{-\beta t}\frac{\partial ^2}{\partial x^2} + \frac{\partial }{\partial x …

In my studies on the Ricci flow, I was faced with a problem. To prove the existence and uniqueness of solutions to the Ricci flow, it is proved that the Ricci flow is a Parabolic P …

It has long been known that the Cauchy initial-value problem for the classical heat equation on $\mathbb{R}$ (or $\mathbb{R}^n$) doesn't have unique solutions, without additional a …

I am looking for solutions of this system. Is such system solvable? here $a,b,c,...$ are smooth functions and $h(x)\in C^1$ and we take $h_i=\frac{\partial \varphi}{\partial x_i}$ …

I am currently studying the paper "CONTINUITY OF SOLUTIONS OF PARABOLIC AND ELLIPTIC EQUATIONS" by John Nash (cf. American Journal of Mathematics, Vol. 80, 1958). The author there …

I have attempted to get an answer on the math.stackexchange but have not got any answer for a while. Thus, I am posting the question here. I am analyzing the following problem in o …

What does the solution $u(x,t)$ of $u_t=u_{xx}+\frac{1}{x}u_x$ on $[0,1]$ with the following initial condition look like? $u(\frac{1}{2},0)=\delta(\frac{1}{2})$ (i.e. delta functi …

I saw several papers that without proof accept the fact "Short time existence on nonlinear parabolic PDE" is there any affirmative proof of this fact? in which book we have this fa …

I am looking for results that show smooth dependence of a solution to a parabolic equation, from the initial data. More specifically I have the following problem: CONSIDER spaces …

I need to know about this non-linear logarithmic fast diffusion equation for a function $u(x,t)$ of one space variable $x$ and time $t$: $$ u_t = (\ln u)_{xx}$$ which is to run on …

I want to learn about parabolic PDE and it seems to me that there is no established reference as far as where one should look if one wants to learn the subject from basics. I thin …

Hi, I have a PDE of the type $$v_t+av_{xx}+bv_{yy}-\gamma v^2_y=0$$ with $a,b,\gamma>0$. The final condition is $$v(T,x,y)=g^n(y)h(x)$$ where $g^n$ is a sequence of smooth functio …

Given a differential equation on a Banach space $\mathcal{X}$ of the form $\frac{d u}{d t} = F(u)$, it is often the case that $F$ is equivariant under translations, i.e. that $T_\a …

How we can prove the short time existence for Kahler Ricci flow on compact manifolds. I know that by linearization and Detork's trick we can prove the short time existence for Ricc …