Exact traveling wave solutions of the stochastic Wick-type fractional Caudrey-Dodd-Gibbon-Sawada-Kotera equation

1.   Introduction

1.1.   Problem statement

We consider a singular no-sign obstacle problem of the type

where $a > 1$, $\chi_{D}$ is the characteristic function of $D$, $B_1\subset\mathbb{R}^{n}$ is the unit ball and $B^{+}_1 = B_1\cap\{x_{1} > 0\}$. The equation is considered in the weak form,

for all $\varphi \in W_0^{1, 2} (B_1^+)$. This problem, when the non-negativity assumption $u \geq 0$ is imposed, is already studied in ^[9]. The above no-sign problem, as a general semilinear PDE with non-monotone r.h.s., introduces certain difficulties and to some extent some challenges. To explain this we shall give a very short review of the existing results and methods for similar type of problems (see also the book ^[6] and Caffarelli's review of the classical obstacle problem ^[2]). The general methodology of approaching such problems lies in using the so-called ACF-monotonicity formula (see ^[8]) or alternatively using John Andersson's dichotomy (see ^[1] or ^[3]). Although there are still some chances that both these methods will work for our problem above, we shall introduce a third method here which relies on a softer version of a monotonicity formula (which has a wider applicability) in combination with some elaborated analysis. We refer to this as a Weiss-type monotonicity formula, see (2.1) below.

1.2.   Notation

For clarity of exposition we shall introduce some notations and definitions here that are used frequently in the paper. Throughout this paper, ${\mathbb R}^n$ will be equipped with the Euclidean inner product $x\cdot y$ and the induced norm $|x|$, $B_r(x_0)$ will denote the open $n$-dimensional ball of center $x_0$, radius $r$ with the boundary $\partial B_r(x_0)$. In addition, $B_r = B_r(0)$ and $\partial B_r = \partial B_r(0)$. ${\mathbb R}^n_+$ stands for half space $\{x\in {\mathbb R}^n: x_1 > 0\}$ as well as $B_r^+ = B_r\cap {\mathbb R}^n_+$. Moreover, in the text we use the $n$-dimensional Hausdorff measure ${\mathcal H}^n$. For a multi-index $\mu = (\mu_1, \cdots, \mu_n)\in {\mathbb Z}^n_+$, we denote the partial derivative with $\partial^\mu u = \partial_{x_1}^{\mu_1}\cdots\partial_{x_n}^{\mu_n}u$ and $|\mu|_1 = \mu_1+\cdots+\mu_n$.

For a domain $\Omega\subset \mathbb{R}^n_+$ and $1\leq p < \infty$, we use the notation ${\mathcal L}^p(\Omega)$ and $W^{m, p}(\Omega)$ as the standard spaces. However, we need some new notation for the weighted spaces

where $\theta\in {\mathbb R}$. For $m\in {\mathbb N}$, we define the weighted Sobolev space $W^{m, p}(\Omega; x_1^\theta)$ as the closure of $C^\infty(\overline\Omega)$ with the following norm,

It is noteworthy that for $\theta = 0$, we have ${\mathcal L}^p(\Omega; 1) = {\mathcal L}^p(\Omega)$ but $W^{m, p}(\Omega; 1)\supsetneq W^{m, p}(\Omega)$. Generally, the trace operator has no sense for $\theta > -1$, while functions in $W^{1, p}(\Omega; x_1^\theta)$ have zero traces on $\{x_1 = 0\}$ for $\theta\leq-1$. (Theorem 6 in ^[7]).

1.3.   Main results

We consider $u\in W^{1, p}(B_1^+, x_1^\theta)$ for some $\theta < -n$ and $n < p$ to be a weak solution of (1.1). This condition provides the continuity of $x_1^{(\theta+n)/p}u$ up to the boundary according to Sobolev embedding Theorem 3.1 in ^[5]. First, we prove the following a priori regularity result.

Proposition 1.1. (Appendix A) Let $u\in W^{1, p}(B_1^+, x_1^\theta)$ be a solution of (1.1) for some $\theta < -n$, $n < p$ and $f\in {\mathcal L}^\infty(B_{1}^+)$. Then for each $\max\{0, 1+\frac{\theta+n}p\} < \beta < 1$ there exists $C = C(\beta, n, a)$ such that for $r\leq1/2$,

for all $x_0\in\{x_1 = 0\}$.

In Appendix A we will prove this proposition. Our main result in this paper concerns the optimal growth rate of solution $u$ of (1.1) at touching free boundary points, which is stated in the following theorem.

Theorem 1.2. Suppose $u\in W^{1, p}(B_1^+, x_1^\theta)$ is a solution of (1.1) for some $\theta < -n$, $n < p$ and $x^0\in \partial\{u = 0\}\cap\{x_1 = 0\}\cap B_{1/4}^+$. Moreover, if $f\in C^{\alpha}(\overline{B_1^+})$ for some $\alpha\in(0, 1)$, then

for a universal constant $C = C(a, n, [f]_{0, \alpha})$.

2.   Monotonicity formula

Our main tool in proving optimal decay for solutions from the free boundary points is Weiss-monotonicity formula, combined with some elaborated techniques. We define the balanced energy functional

Considering the scaling $u_{r, x^0} = u_{r}(x) = \frac{u(rx+x^0)}{r^2}$, $\Phi_{x^0}(r, u) = \Phi_0(1, u_r)$. In what follows we prove almost-monotonicity of the energy.

Lemma 2.1 (Almost-Monotonicity Formula). Let $u$ solve (1.1) and be as in Proposition 1.1 and assume that $\nabla u(x^0) = 0$ for some $x^0\in\{x_1 = 0\}$ and $f\in C^{\alpha}(\overline{B_1^+})$ for some $\alpha\in(0, 1)$. Then $u$ satisfies, for $r\le r_{0}$ such that $B_{r_{0}}^{+}(x^0)\subseteq B_{1}^{+}$,

where $C$ depends only on $\|f\|_{C^{\alpha}(\overline{B_{1}^{+}})}$ and the constant $C(\beta, n, a)$ in Proposition 1.1.

Proof. Let $u_{r}(x) : = \frac{u(rx+x^0)}{r^2}$, then

Note that the second integral

as $|\{u_{r} = 0\wedge \nabla {u_r} \ne 0\}| = 0$ and $\partial_{r}u_{r} = 0$ on $\{u_{r} = 0\wedge \nabla {u_r} = 0\}$. Since $|\partial_{r}u_{r}|\leq Cr^{\beta-2}$ we infer that

and conclude that

Definition 2.2. Let $\mathbb{HP}_{2}$ stand for the class of all two-homogeneous functions $P\in W^{1, 2}(B_{1}^{+}; x_{1}^{a-2})$ satisfying $\operatorname{div}(x_1^a\nabla P) = 0$ in ${\mathbb R}^n_+$ with boundary condition $P = 0$ on $x_1 = 0$. We also define the operator $\Pi(v, r, x^0)$ to be the projection of $v_{r, x^0}$ onto $\mathbb{HP}_{2}$ with respect to the inner product

We will use the following extension of [10,Lemma 4.1].

Lemma 2.3. Assume that $\operatorname{div}(x_1^a\nabla w) = 0$ in $B_1^+$ with boundary condition $w = 0$ on $x_1 = 0$, and $w(0) = |\nabla w(0)| = 0$. Then

and equality implies that $w\in \mathbb{HP}_{2}$, i.e., it is homogeneous of degree two.

Proof. We define an extension of the Almgren frequency,

Moreover, if $N(w, r) = \kappa$ for $\rho < r < \sigma$, it implies that $w$ is homogeneous of degree $\kappa$ in $B_\sigma\setminus B_\rho$.

Now supposing towards a contradiction that $N(w, s) < 2$ for some $s\in(0, 1]$, and defining $w_r(x) : = \frac{w(rx)}{\|w(rx)\|_{ {\mathcal L}^2(\partial B_1^+, x_1^b)}}$, we infer from $N(w, s) < 2$ that $\nabla w_r$ is bounded in ${\mathcal L}^{2}(B_1^+; x_1^a)$ and so $\nabla w_{r_m}\rightharpoonup \nabla w_0$ weakly in ${\mathcal L}^{2}(B_1^+; x_1^a)$ and $w_{r_m} \rightarrow w_0$ strongly in ${\mathcal L}^2(\partial B_1^+; x_1^a)$ as a sequence $r_m \rightarrow0$. Consequently, $w_0$ satisfies $\operatorname{div}(x_1^a\nabla w_0) = 0$ in $B_1^+$, $w_0(0) = |\nabla w_0(0)| = 0$ and $w_0 = 0$ on $x_1 = 0$ as well as $\|w_0\|_{ {\mathcal L}^2(\partial B_1^+; x_1^a)} = 1$. Furthermore, for all $r\in(0, 1)$ we have

and so $w_0$ must be a homogeneous function of degree $\kappa : = N(w, 0+) < 2$. Note that for every multi-index $\mu\in\{0\}\times {\mathbb Z}_+^{n-1}$, the higher order partial derivative $\zeta = \partial^\mu w_0$ satisfies the equation $\operatorname{div}(x_1^a\nabla\zeta) = 0$ in ${\mathbb R}^n_+$. From the integrability and homogeneity we infer that $\partial^\mu w_0\equiv0$ for $\kappa-|\mu|_1 < -\frac n2$, otherwise

can not be bounded. Thus $x'\mapsto w_0(x_1, x')$ is a polynomial, and we can write $w_0(x_1, x') = x_1^ \kappa p(\frac{x'}{x_1})$. Consider the multi-index $\mu$ such that $|\mu|_1 = {\operatorname{deg}} p$, so $\partial^\mu w_0 = x_1^{ \kappa-|\mu|_1}\partial^\mu p$ is a solution of $\operatorname{div}(x_1^a\nabla\zeta) = 0$ in ${\mathbb R}^n_+$. Therefore, $\partial^\mu w_0\in W^{1, 2}(B_1^+; x_1^\theta)$ for $-1 < \theta$ according to Proposition A.1, which implies that $2(\kappa-|\mu|_1)+\theta > -1$. So, ${\operatorname{deg}} p < \kappa+\frac{\theta+1}2$.

Substituting $w_0(x) = x_1^{ \kappa-1}(\alpha x_1+\ell\cdot x')$ for $\kappa > 1$ in the equation and comparing with $w_0(0) = |\nabla w_0(0)| = 0$ we arrive at the only nonzero possible case being $\kappa+a = 2$, which contradicts $a > 1$. The case $\kappa < 1$ leads to ${\operatorname{deg}} p = 0$ and $w_0(x) = \alpha x_1^ \kappa$, which implies $\kappa+a = 1$ and a contradiction to $a > 1$.

3.   Decay rate of solutions close to degenerate points

Proposition 3.1. Let $f\in C^{\alpha}(\overline{B_1^+})$ and $u$ be solution of (1.1) satisfying the condition in Proposition 1.1. Then the function

is bounded on $(0, 1/8)$, uniformly in $x^0\in \partial\{u = 0\}\cap\{x_1 = 0\}\cap B_{1/8}$.

Proof. Let us divide the proof into steps.

Step 1 We claim that there exists a constant $C_1 < \infty$ such that for all $x^0\in \partial\{u = 0\}\cap\{x_1 = 0\}\cap B_{1/8}$ and $r\leq1/8$,

where $u_{x^0, r} : = \frac{u(rx+x^0)}{r^2}$. To prove this we observe that $w : = u_{x^0, r}$ satisfies

Moreover, for $\phi(\rho) : = \rho^{-n-a+1}\int_{\partial B_\rho^+}x_1^{a}w(x)\, d {\mathcal H}^{n-1}$ we have

If $f(x^0)\geq \frac1{8^\alpha}[f]_{0, \alpha}$ then $f_r\geq0$ for $r\leq1/8$. Therefore $\phi$ is increasing and $\phi(\rho)\geq\phi(0) = 0$ (recall that $w(0) = 0$). Similarly, if $f(x^0)\leq -\frac1{8^\alpha}[f]_{0, \alpha}$, we obtain that $\phi(\rho)\leq0$. Therefore the claim is true for $C_1 = 0$ in these cases.

In the case $|f(x^0)|\leq\frac1{8^\alpha}[f]_{0, \alpha}$, then $|f_r(x)|\leq \frac2{8^\alpha}[f]_{0, \alpha}$ and then

So, $|\phi(\rho)|\leq2^{-3\alpha}[f]_{0, \alpha}\rho^2$ and

Step 2 We claim that there exists a constant $C_2 < \infty$ such that

for every $x^0\in\partial\{u = 0\}\cap\{x_1 = 0\}\cap B_{1/8}$, $r\leq1/8$. Suppose towards a contradiction that this is not true, then there exists a sequence $u^m$, $x^{m}\to\bar{x}$ and $r_{m}\to0$ such that

Let $u_{m} : = u_{x^{m}, r_{m}}^m$ and $p_{m} = \Pi(u^m, r_{m}, x^{m})$ and $w_{m} = \frac{u_{m}-p_{m}}{M_{m}}$. Then, since $u_{m}(0) = |\nabla u_m(0)| = 0$ and by the monotonicity formula and the result of previous step, we find that

Passing to a subsequence such that $\nabla w_{m}\rightharpoonup \nabla w$ in ${\mathcal L}^2(B^+_{1};x_1^a)$ as $m\to\infty$, the compact embedding on the boundary implies that $\|w_\|{ {\mathcal L}^{2}(\partial B_{1}\cap \mathbb{R}_+^n; x_1^a)} = 1$, and

and that

Since $\operatorname{div}(x_1^a\nabla w_{m}) = \frac{x_1}{M_{m}}f(x^m+r\cdot)\chi_{\{u_m\neq0\}}$, it follows that $\operatorname{div}(x_1^a\nabla w) = 0$ in $B_{1}^+$. Moreover, we obtain from $L^{p}$-theory that $w_{m}\to w$ in $C_{\text{loc}}^{1, \alpha}(B^+_{1})$ for each $\alpha\in(0, 1)$ as $m\to\infty$. Consequently $w(0) = \lvert\nabla w(0)\rvert = 0$. Thus we can apply Lemma 2.3 and obtain from (3.2) that $w$ is homogeneous of degree 2, contradicting (3.3) and $\|w_\|{ {\mathcal L}^{2}(\partial B_{1})} = 1.$ This proves the claim.

Step 3 We will show that there exists constant $C_2$ such that for all $x^0\in\partial\{u = 0\}\cap\{x_1 = 0\}$ satisfying

we have

In order to see this, we can observe that

Now by condition (3.4), consider a sequence $r_m \rightarrow0$ such that $\frac{|B_{r_m}^+(x^0)\cap\{v = 0\}| }{|B_{r_m}^+|} \rightarrow0$ and assume that $\nabla(u_{x^0, r_m}-p_{x^0, r_m})\rightharpoonup \nabla w$ in ${\mathcal L}^2(B_1^+; x_1^a)$ as $m \rightarrow\infty$. Observe now $\operatorname{div}(x_1^a w) = f(x^0)$ in $B_1^+$ and by similar calculation as above we will have

On the other hand,

Step 4 In this step, we prove the proposition for the points satisfying condition (3.4). For these points, we have

Thus $r\mapsto\partial_{r}\left[\int_{\partial B_{1}^+}x_1^au_{x^0, r}^{2}{\mathop{}\!d} {\mathcal H}^{n-1}\right]$ is integrable and we obtain uniform boundedness of $\int_{\partial B_{1}^+}x_1^au_{x^0, r}^{2}d {\mathcal H}^{n-1} = r^{-n-3-b}\int_{\partial B_{r}(x^0)^+}x_1^au^{2}{\mathop{}\!d} {\mathcal H}^{n-1}$ for all points with property (3.4). It follows that the boundedness holds uniformly on the closure of those points $x^0$.

Step 5 We now consider the case

Let us assume towards a contradiction that there are sequences $u^m$, $r_{m}$ and $x^m$ such that and $M_{m} = \|{u_{x^m, r_m}}\|_{ {\mathcal L}^{2}(\partial B_{1}^+)}\to+\infty$ as $m\to\infty$. Setting $w_{m} = \frac{u^m_{x^m, r_{m}}}{M_{m}}$ we obtain, as in Step 2, that a subsequence of $w_{m}$ converges weakly in $W^{1, 2}(B_{1}^+; x_1^{a-2})$ to a function $w$, with $\|w_\|{ {\mathcal L}^{2}(\partial B_{1}^+; x_1^a)} = 1$, $w(0) = \lvert\nabla w(0)\rvert = 0$, $\operatorname{div}(x_1^a\nabla w) = 0$ and

According to Lemma 2.3, $w\in\mathbb{HP}_2$. In addition we now know that

This however contradicts the analyticity of $w$ inside $B_1^+$, knowing that $\|w_\|{ {\mathcal L}^{2}(\partial B_{1};x_1^a)} = 1$.

Now we are ready to prove the main result of the article.

Proof of Theorem 1.2. From Theorem 8.17 in ^[4], we know that if $\operatorname{div}(b(x)\nabla w) = g$ such that $1\leq b(x)\leq 5^a$, then there exists a universal constant $C = C(a, n)$ such that

Now for $x^0 \in \partial\{u = 0\}\cap\{x_1 = 0\}\cap B_{1/8}^+$ and an arbitrary point $y\in \partial B_r^+(x^0)$, we apply the above estimate for $R = 2\delta/3$, $w = (\delta/3)^au$ and equation $\operatorname{div}\left(b(x)\nabla w\right) = x_1^af\chi_{\{u\neq0\}}$, where $\delta = y_1$ and $b(x) = \frac{x_1^a}{(\delta/3)^a}$. Note that $1\leq b(x)\leq 5^a$ in $B_{2\delta/3}(y)$ and

According to Proposition 3.1,

Hence,

From this theorem it follows that solutions have quadratic growth inside cones.

Corollary 3.2. Suppose $u$ is a solution of (1.1) satisfying the condition in Proposition 1.1 and $x^0\in \partial\{u = 0\}\cap\{x_1 = 0\}\cap B_{1/8}^+$. Then, for every constant $\tau > 0$,

where ${\mathcal C} : = \{x: x_1\geq \tau |x-x^0|\}$.

Acknowledgments

This paper was prepared while M. Fotouhi was visiting KTH Royal Institute of Technology. A. Minne was supported by the Knut and Alice Wallenberg Foundation. H. Shahgholian was supported by Swedish Research Council.

Conflict of interest

The authors declare no conflict of interest.

A.   A priori regularity of the problem

Let $u$ be a solution of (1.1) for $f\in {\mathcal L}^\infty(B_1^+)$. We are going to show a priori regularity for solutions to (1.1). Consider the operator ${\mathcal L}_{a, c}u: = x_1^2\Delta u+ax_1\partial_1u-cu$. The following proposition is the regularity result related to this operator which has been proven by Krylov [5,Theorem 2.7,Theorem 2.8].

Proposition A.1. i) For any $a\in {\mathbb R}, p > 1$ and $\theta\in {\mathbb R}$ there exists a constant $c_0 > 0$ such that for any $c\geq c_0$ the operator ${\mathcal L}_{a, c}$ is a bounded one-to-one operator from $W^{2, p}({\mathbb R}^n_+; x_1^\theta)$ onto ${\mathcal L}^p({\mathbb R}^n_+; x_1^\theta)$ and its inverse is also bounded, in particular for any $u\in W^{2, p}({\mathbb R}^n_+; x_1^\theta)$

where $C$ is independent of $u$ and $c$.

ii) The statement in i) is valid for the operator ${\mathcal L}_{a, 0}$ when $-1 < \theta < a-2$ and $a > 1$ or either $a-2 < \theta < -1$ and $a < 1$.

Now we can deduce a priori regularity result for $u$ as follows.

Proof of Proposition 1.1. Notice that $x_1^{\beta-1}u\in C(\overline{B_1^+})$ due to Sobolev embedding Theorem 3.1 in ^[5]. Then if the statement of proposition fails, there exists a sequence $u_j$ of solutions (1.1), $x^j\in \{x_1 = 0\}$ and $r_j\rightarrow0$ such that

In particular, the function $\tilde u_j(x) = \frac{u_j(x^j+r_jx)}{jr_j^{1+\beta/2}}$, satisfies

and with equality for $R = 1$, along with

where $c_0$ is defined in Proposition A.1. According to (A.1), the right hand side of (A.2) is uniformly bounded in ${\mathcal L}^p(B_R^+; x_1^\theta)$ for $p(\beta-1)-1 < \theta\leq-1$. From here and Proposition A.1 we conclude that $\{\tilde u_j\}$ is bounded in $W^{2, p}(B_{R}^+; x_1^\theta)$ for some $\theta\leq-1$ and there is a convergent subsequence, tending to a function $u_0$ with properties

as well as the condition $\theta\leq-1$ insures that the trace operator is well defined and $u_0$ is zero on $\{x_1 = 0\}$. The Liouville type theorem in [9,Lemma 20]) implies that

Therefore, $u_0$ is a linear function, which contradicts (A.3).