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Blowup and nonexistence of solutions of some semilinear degenerate parabolic equations
Boundary Value Problems volumeÂ 2015, ArticleÂ number:Â 157 (2015)
Abstract
In this paper we study a class of semilinear degenerate parabolic equations arising in mathematical finance and in the theory of diffusion processes. We show that blowup of spatial derivatives of smooth solutions in finite time occurs to initial boundary value problems for a class of degenerate parabolic equations. Furthermore, nonexistence of nontrivial global weak solutions to initial value problems is studied by choosing a special test function. Finally, the phenomenon of blowup is verified by a numerical experiment.
1 Introduction
In this paper, we consider the equation
where \(z= (x,y,t) \) denotes the point in \(\mathbb{R} ^{3}\). This equation arises in mathematical finance [1] and in the physical phenomena such as diffusion and convection of matter. One of the main features of equation (1.1) is the strong degeneracy due to the lack of diffusion in the ydirection. We restrict our consideration to two cases: the initial boundary value problems of (1.1) and the initial value problems of (1.1).
Regarding the theoretical analysis of (1.1), most scholars have been devoted to the study of wellposedness and regularity of solutions [2â€“5]. Antonelli and Pascucci [2] proved that there exists a unique viscosity solution to the initial value problem for (1.1) in a small time. The existence and uniqueness of a global solution in an unbounded domain was studied by Volâ€™pert and Hudjaev [5]. On the regularity of solutions, Citti et al. [3] proved that the viscosity solution of (1.1) is a classical solution in the sense that \(u_{xx} \), \(uu_{y}u_{t} \) are continuous and the equation is pointwise satisfied. Furthermore, they obtained the smooth solution of (1.1) when \(f(z) \in C^{\infty}(\Omega)\) and \(\partial_{x}u \neq0 \), in an open set \(\Omega\subset\mathbb{R}^{3} \) in [4].
Blowup and nonexistence of solutions for (1.1) are as important aspects of properties of partial differential equations. In [6], Fujita described the initial problem of a semilinear parabolic equation, which takes place blowing up even when the initial data is very nice. Ever since then, results about blowup and nonexistence have been generalized to deal with some more general semilinear, quasilinear and fully nonlinear parabolic equations and systems. Without being exhaustive with the amount of references concerned with this topic, let us mention the works [7â€“11]. For a more extensive list of references, we refer to the book by Quittner and Souplet [12].
For degenerate parabolic equations, blowup results have been obtained by many authors, see [13â€“16]. It is mentioned that the initial value problem
has no nontrivial nonnegative solutions in [14]. There is an interesting thing that replacing the right term \(u^{1+\alpha}\) by \(uu^{\alpha}\) in the first equation of the above problem, Haraux and Weissler [17] obtained global solutions.
In this paper, we will mainly deal with the following problems:
and
It is known that the local solutions are obtained for (1.2) and (1.3) in [2]. Our interest is the blowup of spatial derivatives of solutions in finite time to the initial boundary value problem (1.2) and the nonexistence of the weak solutions to the initial value problem (1.3).
Our main results are the following theorems.
Firstly, we define energy functionals
Theorem 1.1
Let \(a_{0}(x)\) have compact support such that \(E(a_{0})<0 \). Assume that the initial value \(g(x,y) \) takes the form \(g=yb_{0}(x,y) \), \(b_{0}(x,0)=a_{0}(x) \). Then spatial derivatives of smooth solutions of (1.2) blow up in finite time. More precisely, there exists \(T=\frac{F(a_{0})}{6(1\beta)E(a_{0})} \), \(\beta\in(1,\frac{3}{2}) \), such that either
This is our first result. A smooth solution u of (1.2) means \(u \in C^{1}([0, T_{0}), C^{2}(\mathbb{R}^{+} \times\mathbb{R})) \) for \(T_{0}>0\). It is remarkable that TheoremÂ 1.1 remains valid if we replace \(uu^{\alpha}\) by \(uu^{\alpha}\) or 0.
Next, we consider the more general case
Continuing with the description of our results, let us introduce the precise assumptions on our f:

(H)
\(f(0)=0\) and there exists an increasing continuous function Ï• on \([0,+\infty)\) such that
$$ \bigl f(r_{1})f(r_{2})\bigr\leq\phi\bigl( r_{1}r_{2}\bigr), $$and \(\frac{1}{\phi(r)} \) is not integrable near \(r = +0 \), that is,
$$ \int_{0}^{\delta} \frac{dr}{\phi(r)} = + \infty, $$where Î´ is a positive constant.
Then TheoremÂ 1.1 can be extended to the following theorem.
Theorem 1.2
Suppose that f satisfies (H), \(f'(0) \leq0 \), and \(g(x,y) \) satisfies the conditions of TheoremÂ 1.1. Then there exist blowup solutions of (1.5) in finite time.
For initial value problems, we derive two theorems.
The following theorem considers blowup of solutions to the initial value problem
Theorem 1.3
Assume that u is the bounded classical solution of (1.6) in \(\overline{Q}_{T_{\varepsilon}} \), \(Q_{T_{\varepsilon}}=\mathbb{R}^{2} \times(0,T\varepsilon) \), for any given \(\varepsilon\in(0,T) \). If \(f\leq0 \) and \(g(x, c_{0}) \geq\frac{c_{0}}{T} \), \(c_{0}>0 \), then u blows up in time T at \(y=c_{0}\).
This improves the result of Example 1.1 in [2].
Finally, we consider nonexistence of weak solutions to the initial problem (1.3). Here, aÂ weak solution of (1.3) is defined as follows.
Definition 1
A function \(u \in L_{\mathrm{loc}}^{2}(Q) \) is called a weak solution of (1.3) with the initial data \(g(x,y) \in L_{\mathrm{loc}}^{1}(\mathbb{R}^{2}) \) in \(Q=\mathbb{R}^{2} \times( 0,\infty)\) if \(t^{k}x^{\gamma}u^{\alpha+1}\in L_{\mathrm{loc}}^{1}(Q) \) and
hold for any nonnegative \(\phi\in C_{0}^{2}(\mathbb{R}^{2}\times [0,\infty))\).
Now, we address our result.
Theorem 1.4
Let \(\alpha>1\), \(k\frac{\gamma}{2}>0\). Assume that \(\int_{\mathbb{R}^{2}}g(x,y)\,dx\,dy \geq0\). If \(\alpha\leq k\frac{\gamma}{2}+1\), then there exists no nontrivial weak solution of (1.3).
The rest of the paper is organized as follows. SectionÂ 2 is devoted to initial boundary value problems (1.2) and (1.5) through energy methods. In SectionÂ 3, we investigate initial value problems (1.3) and (1.6) by a comparison principle and choosing a special test function. Finally, we describe a numerical result about the blowup of solutions in TheoremÂ 1.1 in SectionÂ 4.
2 Initial boundary value problems
In this section, we obtain the blowup results of initial boundary value problems (1.2) and (1.5).
2.1 Proof of Theorem 1.1
Suppose that a smooth solution u of (1.2) exists locally and the initial value \(g(x,y) \) satisfies the form \(g=yb_{0}(x,y) \). If we restrict (1.2) to the half line \(l=\lbrace x>0,y=0 \rbrace\) and let \(v(x,t)=u(x,0,t) \), v obviously satisfies an equation of the form
where \(w(x,t)=u_{y}(x,0,t) \) is smooth, with the initial data \(v(x,0)=0 \) and the boundary data \(v(0,t)=0 \). By the maximum principle, we conclude that \(u(x,0,t)=v(x,t)=0 \) as long as u stays smooth. Any smooth function that vanishes at \(y=0 \) can be written in this form
Substituting (2.1) into the first equation of (1.2), we obtain
Let \(a(x,t)=b(x,0,t) \) and \(a_{0}(x)=b(x,0,0) \). Then a satisfies
with the initial boundary value conditions
The proof of TheoremÂ 1.1 is based on the following lemma.
Lemma 2.1
If \(a_{0}(x) \) has compact support such that \(E( a_{0})<0 \) (E is defined as in (1.4)), then there exists a finite time T such that either
Proof
Assume that \(\max_{x\in\mathbb{R}^{+}}a \) stays bounded. Since a satisfies equation (2.2), the standard result shows that a decays exponentially fast at infinity as long as its maximum norm stays bounded.
Next, we will show that \(F(a) \) (F is defined as in (1.4)) blows up in finite time assuming that \(a_{x}(0,t) \) stays finite. We will use the following integral identities that are valid for the smooth solutions of (2.2)(2.3):
Employing (2.4), (2.5) and (2.6), we find
Thus, we have \(E(a)<0 \) for \(t>0 \) under the condition \(E(a_{0})<0 \).
At last, we compute the time derivative of \(H(a)=\frac{E(a)}{F(a)^{\beta}} \). Firstly, we have
Furthermore,
If we choose \(\beta\in(1,\frac{3}{2}) \), then
By the definition of \(H(a)\), we get \(E(a)\geq H(a_{0})F(a)^{\beta} \), where \(H(a)_{t=0}=H(a_{0})\).
Since
we deduce
Hence there exists a finite time \(T=\frac{F(a_{0})}{6(1\beta)E(a_{0})} \), \(\beta\in(1,\frac{3}{2}) \) such that
Due to the condition
we get
This completes the proof of LemmaÂ 2.1.â€ƒâ–¡
Proof of TheoremÂ 1.1
We note that the smooth solution of (2.2)(2.3) is unique. From LemmaÂ 2.1, we get either
This implies that either
â€ƒâ–¡
2.2 Proof of Theorem 1.2
As the proof of LemmaÂ 2.1, the smooth solution of (1.5) can be written in this form
if \(g=yb_{0}(x,y) \) and f satisfies hypothesis (H). Substituting \(u(x,y,t)=yI(x,y,t) \) into the first equation of (1.5), we get
Let \(s(x,t)=I(x,0,t) \) and \(s_{0}(x)=I(x,0,0) \). By \(f(0)=0 \) and \(f'(0)\leq0 \), multiply (2.7) by \(\frac{1}{y} \) and take limit as \(y\rightarrow0 \) to get
with the initial boundary value conditions
Setting \(\psi=\exp(f'(0)t)s\), Ïˆ satisfies
Lemma 2.2
Define
If the initial value \(\psi_{0}=\psi(x,0) \) has compact support such that \(E_{1}(\psi_{0})<0 \) and \(f'(0)\leq0 \), then there exists a finite time T such that either
Proof
It is proceeded by a contradiction to the proof of LemmaÂ 2.1, that is, we assume that \(\psi_{x}(0,t) \) stays finite and we will get \(F(\psi) \) (F is defined as in (1.4)) blows up in finite time. The following integral identities are valid for smooth solutions of (2.8):
Due to \(f'(0)\leq0 \), we get
Since
we have
If we define \(H(a)=\frac{E_{1}(a)}{F(a)^{\beta}} \) and choose \(\beta\in(1,\frac{3}{2}) \), then
We have \(E_{1}(\psi)\geq H(\psi_{0})F^{\beta} \) and \(\frac{dF}{dt}\geq6E_{1}\geq6H(\psi_{0})F^{\beta}\), where \(H(\psi)_{t=0}=H(\psi_{0})\). Hence there exists a finite time \(T=\frac{F(\psi_{0})}{6(1\beta)E_{1}(\psi_{0})} \), \(\beta\in(1,\frac{3}{2}) \) such that
This completes the proof of LemmaÂ 2.2.â€ƒâ–¡
By LemmaÂ 2.2, the solution of (1.5) has either
Then TheoremÂ 1.2 is obtained.
Remark 1
Replacing the semilinear term \(uu_{y}\) of (1.5) by \(h(u)u_{y}\), if \(h(u) \) satisfies hypothesis (H) and \(f'(0)h'(0) \leq0 \), then the smooth solutions of (1.5) have the same result as TheoremÂ 1.2.
Remark 2
TheoremÂ 1.1 and TheoremÂ 1.2 describe the lower dimensional problems. The higher dimensional cases are parallel to the lower dimensional cases. For example, the high dimensional problem is as follows:
where \(\Delta_{x} \) is the Laplace operator acting in the variable \(x=(x_{1}, x_{2}, \ldots, x_{N} ) \in \mathbb {R}^{N}_{+} \).
3 Initial value problems
The section describes initial value problems (1.3) and (1.6) for deriving the proofs of TheoremÂ 1.3 and TheoremÂ 1.4.
For the convenience of description, we set
Next, we get a comparison principle about the initial value problem (1.6).
Lemma 3.1
Assume that there are two solutions \(u_{i}\) of (1.6) satisfying \(u_{i} \in C^{2,1}(Q_{T}) \cap C(\overline{Q}_{T}) \) and \(u_{i} , (u_{2})_{y} \in L^{\infty}(Q_{T}) \), \(i=1,2 \). Let \(f(u_{1}) \leq f(u_{2}) \), \(g(u_{1}) \geq g(u_{2}) \), then \(u_{1} \geq u_{2} \).
Proof
Set \(w=u_{1}u_{2} \),
We suppose that \(r_{0} >0\), \(\alpha>0 \), \(N>0 \), and
and set
for \(r^{2}=x^{2}+y^{2} \).
Defining LÌ… by
we have
Choosing
we get \(\overline{L}v \leq0 \).
In \(\Omega_{r_{0}}=\{(x,y,t)x^{2}+y^{2}\leq r_{0}^{2},0\leq t\leq T\} \), due to \(v_{t=0} \geq0 \), \(v_{r=r_{0}} \geq0 \), by the maximum principle, we obtain \(v\geq0 \).
For any \(p \in Q_{T} \), if we choose \(r_{0} \) sufficiently large such that \(p \in\Omega_{r_{0}} \), then \(v _{p}\geq0 \).
Set \(r_{0}\rightarrow\infty\), we get \(w _{p}=(u_{1}u_{2}) _{p}\geq0 \).â€ƒâ–¡
Using LemmaÂ 3.1, we get the following proof of TheoremÂ 1.3.
Proof of TheoremÂ 1.3
Taking \(u_{1}=\frac{y}{Tt} \), it shows that
Fixing \(y=c_{0}>0 \), we have \(Lu_{1}(x,c_{0},t)\geq Lu(x,c_{0},t) \). When \(g(x,c_{0}) \geq\frac{c_{0}}{T} \), we get \(u\geq \frac{c_{0}}{Tt} \) by LemmaÂ 3.1.
At \(y=c_{0} \),
â€ƒâ–¡
Finally, we give the proof of TheoremÂ 1.4.
Proof of TheoremÂ 1.4
Let u be such a weak solution of (1.3) and \(\phi\in C_{0}^{2}(\mathbb {R}^{2}\times [0,\infty))\) be a nonnegative test function. Applying the first equation of (1.3) and Youngâ€™s inequality, we obtain
where \(\alpha>1\).
We define
where \(\psi\in C_{0}^{\infty}(\mathbb {R}^{+})\) satisfies \(0\leq\psi\leq1 \) and
Then
In the case where \(\gamma+2\alpha2k2<0\), the exponents of the right terms in (3.1) are negative. Taking the limit as \(r \rightarrow\infty\) in (3.1), we deduce that
This implies that \(u\equiv0\) in Q.
In the case where \(\gamma+2\alpha2k2=0\), we get from (3.1) that
Set \(\Omega_{r}=\{(x, y, t)\in\mathbb{R}^{2}\times(0,\infty):r^{2} \leq t+x^{2}+y^{2} \leq2r^{2}\}\). Since \(\psi(s)\) is constant for \(s\in[0,1]\cup[2,\infty)\), we have
It follows from the integrability of \(t^{k}x^{\gamma}u^{\alpha+1} \) in Q that
From (3.1) and (3.2), we know that
This implies that \(u\equiv0\).â€ƒâ–¡
4 A numerical experiment
Next, we present a numerical experiment. Our goal is to show that the result presented in TheoremÂ 1.1 can be observed when one performs numerical computations. For a numerical experiment, we choose an adaptive bounded space to problem (1.2).
At \(y=0\), (1.2) in a bounded domain can be written to the following problem:
FigureÂ 1 shows the evolution of the numerical solution of (4.1) with a space step size 0.01, whose blowup time turns out to be \(T = 0.56\). In \((0,T) \), for any \(T \geq0.56 \), we fail to show the figure in Matlab since the function value increases rapidly. In FigureÂ 2, we display the profile of FigureÂ 1 at \(t=0.55\).
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Acknowledgements
This work was done when the author was visiting the Institute of Mathematical Sciences, the Chinese University of Hong Kong. The author would like to express her sincere thanks to Professor Zhouping Xin for his helpful references and fruitful comments. The author also would like to express her deep gratitude to the anonymous referee for careful reading and valuable suggestions. The author is supported by the Research Innovative Program of Jiangsu Province (No. CXLX13188) and the Excellent Ph.D Student Foundation of NUST.
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Wu, H. Blowup and nonexistence of solutions of some semilinear degenerate parabolic equations. Bound Value Probl 2015, 157 (2015). https://doi.org/10.1186/s1366101504234
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DOI: https://doi.org/10.1186/s1366101504234