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.

  • $\begingroup$ Is $\Delta$ the standard Laplacian on $\mathbb{R}$? $\endgroup$ – Deane Yang Jul 22 '13 at 14:04
  • $\begingroup$ with $n=1$ yes. $\endgroup$ – JCM Jul 22 '13 at 14:08
  • $\begingroup$ Sorry, I intended to only have the nonlinear term depending on $u$ (I suppose that it could depend on $t$ too, but most certainly not on $x$). $\endgroup$ – JCM Jul 22 '13 at 15:28
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    $\begingroup$ You cannot have spatially inhomogeneous solutions if you have uniqueness, because any translate of a solution is also a solution. Since uniqueness holds under quite mild assumptions, I doubt your claim that you have constructed spatially inhomogeneous solutions. At least you should explain more about how this is possible and what is different from "standard" situations. $\endgroup$ – Michael Renardy Jul 22 '13 at 19:01
  • $\begingroup$ Do you choose $f$ to have particularly low regularity? $\endgroup$ – Willie Wong Jul 23 '13 at 9:50

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

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

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

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  • $\begingroup$ Good suggestions. $\endgroup$ – JCM Jul 23 '13 at 11:58
  • $\begingroup$ I should also say thankyou. I was aware of the Haraux and Weissler paper (probably the closest thing I have seen to the questions I have considered), albeit as you mention it does have a slightly different flavour ($L_p$ setting). Nonetheless, I can continue my reference search now. $\endgroup$ – JCM Jul 23 '13 at 12:13
  • $\begingroup$ I should also add that in the solution setting I am interested in (see the word bounded), locally Lipschitz nonlinearities are sufficient for uniqueness. Of course as you have pointed out in the references above, in more general solution settings this is not true. $\endgroup$ – JCM Jul 24 '13 at 9:57

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.

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    $\begingroup$ This is merely an existence result, no guarantee is given with regard to spatial inhomogeneity. For all we know, the contraction mapping based existence result will just give solutions to the first order ode $u_t=f(u)$ with $u(0)=0$. $\endgroup$ – JCM Jul 22 '13 at 18:37
  • $\begingroup$ I should also comment that I have already constructed spatially inhomogeneous solutions to these type of problems. I am merely looking for references for other works which have (if they exist) also bothered to do so ... and maybe find out why they bothered to construct them too. $\endgroup$ – JCM Jul 22 '13 at 18:39
  • $\begingroup$ Could you provide explicit examples of spatially inhomogeneous solutions to your problem? $\endgroup$ – Deane Yang Jul 23 '13 at 3:04
  • $\begingroup$ I could but it took me (for the first type of result) about 10 pages to construct (the second took a further 10), so perhaps not here. I will happily send you a copy when it is all written up (in a slightly more presentable form) though. $\endgroup$ – JCM Jul 23 '13 at 11:06
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    $\begingroup$ Sorry a response took so long ... I forgot my login details ... see arxiv.org/abs/1607.08423 $\endgroup$ – JCM Sep 20 '18 at 1:25

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