Reference Request: Spatially inhomogeneous solutions to parabolic PDE with homogeneous initial data

Question

I am interested in spatially inhomogeneous classical bounded solutions $u:\mathbb{R}^n \times [0,T] \to \mathbb{R}$ to the Cauchy problem for semi-linear parabolic PDE, which have homogeneous initial data, i.e; $$ u_t - \bigtriangleup u = f(u) \ \ \ \forall (x,t)\in\mathbb{R}^n\times (0,T] $$ $$ u(x,0)= 0 \ \ \ \forall x\in\mathbb{R}^n .$$ If anyone has any references to similar works on this type of problem (specifically concerning spatially inhomogeneous solutions), I would be most appreciative.

Note that the question of when solutions will be spatially homogeneous (given conditions on $f$) is not of interest to me as it is besides the point. The reason I obtained this result was simply because it seemed somewhat counter-intuitive to most peoples (and initially my own) understanding of these type of problems.

Answer 1

While not exactly the same problem that you solved, there has been previous results considering nonuniqueness of solutions (with zero initial data) for power-law type semilinear term. Interestingly, contrary to what you wrote, Lipschitz may not be enough (depending on the function spaces in consideration) for uniqueness.

Some relevant papers: In the case where the nonlinearity is Lipschitz and the function spaces used are $L^p$ type spaces, we have

• Haraux and Weissler. "Nonuniqueness for a semilinear initial value problem". http://www.ams.org/mathscinet-getitem?mr=648169
• Ni and Sacks. "Singular behavior in nonlinear parabolic equations". http://www.ams.org/mathscinet-getitem?mr=768731
• Baras. "Non-unicité des solutions d'une équation d'évolution non-linéaire". http://www.numdam.org/item?id=AFST_1983_5_5_3-4_287_0

In the case where the nonlinearity is not Lipschitz, we have

• Fujita and Watanabe, "On the uniqueness and non-uniqueness of solutions of initial value problems for some quasi-linear parabolic equations". http://www.ams.org/mathscinet-getitem?mr=234129

This should be enough to get you started with the literature search on MathSciNet.

Answer 2

If you convolve the equation with the heat kernel in both the $t$ and $x$ variables, you get an equation of the form $$ u = H*f(u). $$ You can then solve this using a contraction mapping or iteration argument using an appropriate norm on $u$ and for sufficiently small $T$. This will give a solution $u$ that decays at infinity (this will be implied by the norm you use) and is smooth for positive $t$ (assuming that $f$ is a smooth function of $u$). I hope someone can provide a specific reference where this is carried out in detail.

EDIT: I didn't read or think about the question carefully enough. in particular, I didn't see "spatially inhomogeneous".

And Michael Renardy is right. It appears to me that for any space of functions $u$ where $H*f(u)$ lies in the same space, there is uniqueness and therefore $u$ is the solution to the ODE.