Suppose we have the following one-dimensional generalized heat equation:
$$u_t(x,t)=g(x,t)\Delta u(x,t), \quad x\in \mathbb{R},t\in(0,\infty).$$
I need to prove that this equation is ill-posed, for some initial data and some particular $g(x,t)$. Is there any literature on these equations? I have found loads on equations like $$u_t=Δu(x,t)+f(x,t)$$
with different $f(x,t)$, but nothing where the Laplacian is multiplied by another function.
Thanks a lot guys!
Let me write an answer which is in fact too long for a comment: the heat equation itself is ill-posed locally in space. Consider the fundamental solution of $\partial _t-∆_x$ ($t$ is the time variable, $x$ is the space variable in $\mathbb R^d $)$$ E(t,x)=H(t)(4π t)^{-d/2}\exp{-\frac{\vert x\vert^2}{4t}}. $$ That function is locally integrable (is also a tempered distribution), with support $\{t\ge 0\}$ and is $C^\infty$ on $(\mathbb R_t\times\mathbb R^d_x)\backslash\{(0,0)\}$. As a result we have for $x_0\not=0$, a smooth function $E$ such that $$ \partial _t E-∆_x E=0,\quad\text{on $\{(t,x)\in \mathbb R\times B(x_0,\vert x_0\vert)\}$},\quad \text{supp } E\subset\{t\ge 0\}, $$ violating unique continuation locally.
If you require the equation to be globally satisfied wrt the space variable, you do have well-posedness results for the heat equation. However the above example shows that you should be more precise about the notion of well-podedness that you want to use, in particular with respect to localization.
If your particular function $g(x,t)$ is identically $-1$, then it becomes $$u_t(x,t)+\Delta u(x,t) =0\ \ \ x∈ℝ,t∈(0,∞)$$ which is equivalent to the backward heat equation. And it is known that this is not in general well-posed.
Sincerely,
If $g(t,x)$ is bounded smooth and say $g(t,x)\le-1$ on a strip $0\le t\le T$, then one could solve the problem backwards starting from $t=T$ with rough data. By general properties of parabolic equations the solution becomes smooth for $t<T$. Reversing time in the usual direction, you have constructed smooth data at $t=0$ and a solution which becomes rough at $t=T$.
This answers in a rather weak sense your question; I guess you are interested in stronger forms of ill-posedness. Probably playing with the previous idea (e.g., repeating at a sequence of times $T_n\to0$ and summing the resulting solutions) it should be possible to produce reasonable data for which no local smooth solution exists, but one should check the details.
The semi-linear case $\partial_tu+Lu=f(u,\nabla u)$, where $L$ is an elliptic linear operator (like the Laplacian $-\Delta$), has a rather simple philosophy. One follows the guidelines of the Cauchy-Lipschitz theory for ODEs. If $a\in X$ is the initial data, one rewrites the Cauchy problem as an integral equation $$u(t)=e^{-tL}a+\int_0^t e^{(s-t)L}f(u(s),\nabla u(s))ds=:Nu(t).$$ When $T>0$ is small enough and the Banach space $X$ is appropriate, one proves that $N$ is a contraction in some ball $B(a;r)$ of $C(0,T;X)$. Then Picard's theorem tells that there is a unique fixed point $u$ ; this is the local solution.
By appropriate, I mean that the operators $S_t:=e^{-tL}$ form a strongly continuous semi-group over $X$. In particular $S_t\in{\mathcal L}(X)$ and $$\|S_t\|_{{\mathcal L}(X)}\le Ce^{\omega t},\qquad\forall t>0$$ for suitable constants $C$ and $\omega$. In practice, we may deal with a scale of Banach spaces $X_s$, like the Sobolev spaces $H^s$, and we have $$\|S_t\|_{{\mathcal L}(X_s,X_r)}\le Ct^{\alpha(s-r)}e^{\omega t},\qquad\forall t>0$$ for some $\alpha>0$, which is related to the order of $L$.
