Suppose we have the following one-dimensional generalized heat equation:

$$u_t(x,t)=g(x,t)\Delta u(x,t) \ \ \ x∈ℝ,t∈(0,∞)$$

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 such equation? I have found loads on equations like $$u_t=Δu(x,t)+f(x,t)$$

with different  $f(x,t)$, but no one where the Laplacian is multiplied by another function.

Thanks a lot guys!

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!

Suppose we have the following one dimensional generalized heat equation:

$$u_t(x,t)=g(x,t)\Delta u(x,t) \ \ \ x∈ℝ,t∈(0,∞)$$

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 such equation? I have found loads on equation like $$u_t=Δu(x,t)+f(x,t)$$

which different $ \ f(x,t)$, but no one where the Laplacian is multiplied by another function.

Thanks a lot guys!

Suppose we have the following one-dimensional generalized heat equation:

$$u_t(x,t)=g(x,t)\Delta u(x,t) \ \ \ x∈ℝ,t∈(0,∞)$$

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 such equation? I have found loads on equations like $$u_t=Δu(x,t)+f(x,t)$$

with different $f(x,t)$, but no one where the Laplacian is multiplied by another function.

Thanks a lot guys!