Paweł Strzelecki | Problems, set no. 4

Here is the fourth set of problems for you.

Problem 1. Consider the heat equation $u_{t} = u_{xx}$ in $\mathbb{R}\times\mathbb{R}$ , with an initial condition

$u(x,0) = \frac 1{1+x^2}\, , \qquad x\in \mathbb{R}.$

Prove that this problem has no analytic solution (i.e., the Taylor series of a solution converges only at $(0,0)$ .)

(This example is due to Kowalewska.)

If you solve all the following problems, you will know that $-\Delta u = f(u)$ has a nontrivial solution in $W^{1,2}_0(\Omega)$ for a large class of nonlinear $f$ ’s.

Here are the assumptions and notation:

(1) $f$ is $C^1$ on $\mathbb{R}$ and satisfies two growth conditions: $\vert f(z) \vert \le C(1+\vert z\vert^p)$ and $\vert f'(z) \vert \le C(1+\vert z\vert^{p-1})$ for some $1 < p < (n+2)/(n-2)$ ;

(2) $F$ is the primitive function of $f$ such that $F(0)=0$ ; we also assume that for some $\gamma < 1/2$ the following holds:

$0 \le F(z) \le \gamma z f(z), \qquad z\in \mathbb{R},$

and $a\vert z\vert^{p+1}\le \vert F(z)\vert \le A \vert z\vert ^{p+1}$ for some $A\ge a> 0$ ;

(3) $H$ stands for $W^{1,2}_0(\Omega)$ with the scalar product

$(u,v)=\int_{\Omega} Du\cdot Dv\, dx\, ,$

and $\Omega \subset\mathbb{R}^n$ is bounded and smooth;

(4) Finally, we define $I\colon H\to \mathbb{R}$ by

$I(u) = \frac 12 \int_{\Omega} \vert Du\vert ^2 dx - \int_{\Omega} F(u)\, dx =: I_{1}(u) - I_{2}(u)\, .$

Now, the problems to solve. (You may wish to consult the proof of Theorem 3 in Evans, Section 8.5.2, as you go through them.)

Problem 2. Prove that for $u\in H$ we have $f(u)\in L^q$ for $q=2n/(n+2)$ ; hence, $I$ is well defined on $H$ . (Use Sobolev imbedding theorem).

Problem 3. Write $H^{-1}(\Omega)$ to denote the space of continuous linear functionals on $H$ . Prove that, for each $u\in H$ , the function $f(u)$ defines a functional $\phi\in H^{-1}(\Omega)$ via

$\phi (v) = \int_{\Omega} f(u)v\, dx\, , \qquad v \in H\, .$

(A comment: people, including the owner of this page, often abuse notation and write things like $f(u)\in H^{-1}(\Omega)$ or $L^{2n/(n+2)}\subset H^{-1}$ etc.)

Problem 4. Prove that for each $w\in H^{-1}(\Omega)$ the equation $-\Delta v = w$ has a unique weak solution $-v\in W^{1,2}_0(\Omega)$ . (Hint: recall Lax–Milgram; Evans, Section 6.2.1.)

Check that the map

$K\colon H^{-1}(\Omega) \ni w \longmapsto v \in W^{1,2}_0(\Omega)$

is an isometry. (Do not be afraid: this is true practically by definition, no hard computations required.)

Problem 5. Use the definition of the derivative (of a function on a Banach space) and check that $I_{1}'(u)=u$ and $I_{2}'(u)=K(f(u))$ for all $u\in H$ . Further, use the growth condtions for $f$ to check that $I_k'$ is Lipschitz on bounded subsets of $H$ , $k=1,2$ .

Problem 6. Use assumptions on $F$ and Rellich-Kondrashov to prove that $I$ satisfies the Palais–Smale condition on $H$ :

(PS) if the sequence $I(u_k)$ is bounded and $I'(u_k)\to 0$ , then a subsequence of $u_k$ converges to some $u\in H$ .

(Hint: Rellich-Kondrashov allows to identify the limit $u$ of $u_k$ ; then, one has to rely on the results of Problems 4-5.)

Problem 7. Check that $I(u)$ is positive on small spheres $\mathbb{S}(0,r)\subset H$ (use Sobolev imbedding again) and negative for $\| u \|> R$ for large $R$ (for this, consider the behaviour of $I$ on rays: analyze $g(t)=I(tu)$ for $t>0$ large).

Once you do all that, use Mountain Pass Theorem to conclude immediately that $-\Delta u = f(u)$ has a nonzero solution with zero Dirichlet boundary conditions.