0
$\begingroup$

I'm reading Avner Friedman, Partial Differential Equations of Parabolic Type. Let \begin{eqnarray} L:=a_{ij}\dfrac{\partial}{\partial x_i\partial x_j}+b_i\frac{\partial}{\partial x_i}+c-\frac{\partial}{\partial t} \end{eqnarray} be a uniformly parabolic differential operator. Is the fundamental solution of equation $Lu=0$ unique under the following condition?

Let $D$ be a domain in $\mathbb{R}^n$, $T_0<T_1, 0<\alpha<1$. $\Omega:=\bar{D}\times[T_0,T_1]$. $a_{ij}, b_i, c:\Omega\to\mathbb{R}$ are bounded and $\exists A>0$ s.t. $\forall x,y\in D, t,s\in[T_0,T_1]$ \begin{eqnarray} |a_{ij}(y,s)-a_{ij}(x,t)|\leq A\left(|y-x|+\sqrt{|s-t|}\right)^\alpha \\ |b_i(y,t)-b_i(x,t)|\leq A|y-x|^\alpha \\ |c(y,t)-c(x,t)|\leq A|y-x|^\alpha \end{eqnarray}

$\endgroup$
0
$\begingroup$

If $LF=\delta$ and $Lu=0\,,$ then $L(F+u)=\delta\,,$ and it would seem that $L$ has nonzero kernel. Are there conditions on $F$?

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.