Abstract
We focus on the nonexistence of global weak solutions of nonlinear Keldysh type equation with one derivative term. In terms of the analysis of the first Fourier coefficient, we show the solution of singular initial value problem and singular initial-boundary value problem of the nonlinear equation with positive initial data blow-up in some finite time interval.
1. Introduction
In this paper we consider the nonexistence of global weak solution of nonlinear Keldysh type equation (1) with modified initial data (2) and boundary value (3); that is,andwhere is the Laplace differential operator and is a parameter. Here (1)-(2) is a singular initial value problem for . And (1)-(3) is a singular initial-boundary value problem when is a bounded domain with a piecewise smooth boundary . The singularity of the given second datum was analyzed by T. Hristov in [1]. In order to formulate the mechanism of blow-up, we assume that, for with a positive constant , the nonlinear source term is a convex function such thatwith some positive constants and .
The Keldysh type equations are known in some specific applications in plasma physics, optics, and analysis on projective spaces [2]. In one-dimensional space, homogeneous equation of (1) turns into the classic Keldysh equation which is closely connected with flow dynamics [3, 4]. In this case, Chen S.X. [5] established the fundamental solutions to the corresponding differential operator for by using the finite part of divergence integrals in theory of distribution. For , the existence of singular solution to Protter problem and Protter-Morawetz problem was considered in [6, 7].
In the upper half space, (1) is a hyperbolic equation with variable propagation speed and characteristics degenerate on . For nonlinear wave equation with constant coefficients, there exist many well-known results including existence and blow-up of solutions. For the existence of solutions, one can see [8–11]. Here we emphasize the nonexistence of global solution; then we will enumerate some classical results. In [12] for , Keller used a comparison theorem and Riemann function showed the solution is blow-up in finite time. In [13], Jörgens extended Keller’s idea to the case for nonlinear wave equation with . For more results, one can see [14–18]. Similar idea of the proof also has been used to establish the blow-up of solution of nonlinear parabolic equations by Kaplan [19] and Fujita [20, 21].
In this paper, we use the first Fourier coefficient of the solution to show the blow-up of solution. This method was first introduced by Kaplan [19] to study a nonlinear second-order parabolic equation, later modified by Glassey [14] for classical solutions of nonlinear wave equations and extended by Levine [22, 23] to weak solutions of parabolic and wave equations. In order to show the first Fourier coefficient of the solution is positive and approaches infinity at finite time, we derive a nonlinear second-order singular differential inequality:Then, based on the cautious calculation of singular differential inequality (5), we can deal with the nonexistence of global solution to the nonlinear problem and the proof is independent of the weak conservation of energy law and concavity of argument.
This paper is organized as follows: In Section 2, the blow-up result for singular initial-boundary value problem is obtained. In Section 3, based on a similar method used in Section 2 but with some differences in calculus, we show the blow-up of solution to singular initial value problem.
2. Singular Initial-Boundary Value Problem
In this section, we consider the blow-up of solution of problem (1)-(3) on a bounded domain . The method of proof is independent of the spatial dimension and the Riemann function.
Let denote the first eigenfunction of Dirichlet problem:with the first eigenvalue . According to a classical result ([24], Vol.I, pp. 451-455), then there is only one normalized eigenfunction satisfying
Now we consider problem (1)-(3) in bounded domain with smooth boundary .
Assume that(a), for all ; there exist two points , , such that , ;(b), .
Set
Definition 1. The function is called a weak solution of initial-boundary value problem (1)-(3) if and are continuous with respect to , if u satisfies (2) and (3), and if for all which are twice continuously differentiable and satisfies
Theorem 2. Assume that a weak solution of problem (1)-(2) with in (4) under assumptions (a)-(b) in the sense of Definition 1 exists; then there is some positive constant such that
Proof. For defined in (6)-(7), setand then andwith . Since there is no differentiability of second order on the weak solution with respect to in Definition 1, then we will derive the second order of by use of (9). Multiplying on (13), in terms of (8) and (9), we obtainAccording to the convexity of and Jensen’s inequality, one hasThen substitute (15) into (14), together with (4) for ; we obtainwhich is equivalent toNote that and ; then there exists a positive number such that for . Hence (17) means that is strictly increasing on the interval . Multiply on (15) and then integrate it over such that ,It follows that and on the entire existence interval .
Now, for , we setOn one hand, it is easy to verify that holds and , . For , integrating by parts, it follows thatOn the other hand, in terms of (17)-(20), we derive thatThen, substituting (21) into (22), we obtainwhich is equivalent toFor , integrate inequality (24) over ; thenFix ; inequality (25) implies that there is a singularity of at some point , whereThen, for some , we deriveMoreover, by a direct computation on (21), we haveHence, combining (20) through (27) and (28), we confirm thatFinally, because of , in terms of Hölder’s inequality, we havewhich implies (11). In particular, since satisfies (7), then, from (30), we obtainThis derives (10).
Next, we consider the case for . Therefore . Then, for , we have . Since , , , and on the entire existence interval from (18), it follows thatThis is equivalent toNote is increasing; thenFix and for and integrate (34) on ; it follows thatSet ; then (35) implies that there exists some point such thatHence, there is such that (30) holds for . Then (10) and (11) are yielded by use of Hlder’s inequality.
Finally, we complete the proof of Theorem 2.
3. Singular Initial Value Problem
In this section we show the blow-up of solution of singular initial value problem (1)-(2) with the idea of proof is similar to that used in Section 2, but it is more complicated in calculus.
Let denote the first eigenfunction of Dirichlet problem (6); then there is only one eigenfunction satisfying (7). As shown in [24], set in (6) and denote as the eigenfunction corresponding to the first eigenvalue ; then is also the function of only and decrease with .
Assume that(c)nonnegative functions , have support in for some positive constant and ; there exists such that and ;(d), .
Set
Taking a similar procedure of the proof of and in [25] with the principle of superposition, we can obtain a solution of problem (1)-(2) which is the representation at a point as a solution of the homogeneous linear equation plus a Volterra integral (formed by confluent hypergeometric function; see [26]) of the inhomogeneous term against an appropriate kernel over the retrograde cone through . Then we give the following definition which is similar to definition 3 in [23].
Definition 3. One says that a solution of problem (1)-(2) has a cone compact support property if (for any ) whenever has support in , has support in for all , where
Now fix and the value will be given later. Let ; then given in Definition 3 has support in , which is contained in .
Definition 4. The function is called a weak solution of problem (1)-(2) which enjoys the cone compact support property, if and are continuous with respect to , if u satisfies (2), and if for all which are twice continuously differentiable and satisfies
Theorem 5. Assume that a weak solution of problem (1)-(2) with in (4) under conditions (c)-(d) in the sense of Definition 4 exists; then there is some positive constant such that
Proof. Define a function ; that is,where is first eigenfunction of Dirichlet problem (6); is a twice continuously differentiable function of which satisfies and for . When , substitute by in (39); then it is easy to verify thatsince the support of is contained in .
For , setand then, by the assumption,In terms of (43) and (46), it follows thatTherefore, we obtainwhich is equivalent toIn terms of the convexity of and Jensen’s inequality, one hasThen substituting (50) into (49), together with (4) for , we obtainIn fact, if set , then we can choose such that for Dirichlet problem (6) in , and then (51) holds which is independent of (4).
Note that and ; then there exists an interval such that for . Hence, on ,This means that is strictly increasing on . Then we obtain on . Multiply on (51) and then integrate it over where :It follows that and on the entire existence interval.
Now, for , we setThen, there, holds and , . For , integrating by parts, it follows thatCombining to (51)-(56), we derive thatwhich is equivalent toThen, for , integrating (58) over , we obtainAccording to the previous analysis, we have independent of ; thenSet , in terms of (59)-(60); then is a finite positive number independent of and , andNow fix ; we choose ,and then we obtainfor all , where is the existence interval of solution (1)-(2). Hence there is a , such thatFrom (55)-(56) and (64), we obtainAt last, by use of Hölder inequality on (65), we derive (40) and (41) for .
For , we also obtain (35) with the term being substituted by by a procedure as mentioned in Section 2; that is,Since we do not know the sign of the term , we calculatewhere we use and .
Let which is a finite positive constant independent of and ; thenFixing , we choose :and then it follows from (68) thatfor all , where is the existence interval of solution (1)-(2). Hence there is , such thatBy a computation with (55)-(56) and (71), we obtainHence, in terms of Hölder inequality, the last formula implies (40) and (41) for .
Then we conclude that Theorem 5 holds.
Data Availability
No data were used to support this study.
Conflicts of Interest
The author declares that there are no conflicts of interest regarding the publication of this paper.
Acknowledgments
This work was supported by NNSF of China (Grant no. 11326152), NSF of Jiangsu Province of China (Grant no. BK20130736), and NSF of Nanjing Institute of Technology (CKJB201709).