Printer Friendly

Nonexistence of Global Weak Solutions of Nonlinear Keldysh Type Equation with One Derivative Term.

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,

t[[partial derivative].sup.2.sub.t]u - [DELTA]u + b[[partial derivative].sub.t]u = f (u), in (0, T) x [OMEGA]

u (0, x) = [[phi].sub.1] (x), (1)

[mathematical expression not reproducible] (2)

and

u(t, x) = 0 on (0, T) x [partial derivative][OMEGA] (3)

where [mathematical expression not reproducible] is the Laplace differential operator and b [member of] (0, 1) is a parameter. Here (1)-(2) is a singular initial value problem for [OMEGA] = [R.sup.n]. And (1)-(3) is a singular initial-boundary value problem when [OMEGA] [subset or equal to] [R.sup.n] is a bounded domain with a piecewise smooth boundary [partial derivative][OMEGA]. The singularity of the given second datum was analyzed by T. Hristovin [1]. In order to formulate the mechanism of blow-up, we assume that, for s > [s.sub.0] with a positive constant [s.sub.0], the nonlinear source term f(s) is a convex function such that

f (s) - [mu]s [greater than or equal t] [cs.sup.1+p], p > 0 (4)

with some positive constants [mu] and c.

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 b < 1/2 by using the finite part of divergence integrals in theory of distribution. For n = 3, 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 t = 0. 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 n = 3, Keller used a comparison theorem and Riemann function showed the solution is blow-up in finite time. In [13], Jorgens extended Keller's idea to the case for nonlinear wave equation with n > 3. 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:

d/dt ([t.sup.b]F' (t)) [greater than or equal to] [ct.sup.b-1] [F.sup.1+p] (t). (5)

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 [OMEGA]. The method of proof is independent of the spatial dimension and the Riemann function.

Let [psi](x) denote the first eigenfunction of Dirichlet problem:

[DELTA][psi] + [lambda][psi] = 0, in [OMEGA]

[psi] (x) = 0 on [partial derivative][OMEGA] (6)

with the first eigenvalue [lambda]. According to a classical result ([24], Vol.I, pp. 451-455), then there is only one normalized eigenfunction [psi](x) > 0 satisfying

[[integral].sub.[OMEGA]] [psi](x) dx = 1. (7)

Now we consider problem (1)-(3) in bounded domain Q with smooth boundary [partial derivative][OMEGA].

Assume that

(a) [[phi].sub.1](x) [greater than or equal to] 0, [[phi].sub.2](x) [greater than or equal to] 0 for all x [member of] [OMEGA]; there exist two points [x.sub.i] [member of] [OMEGA], i = 1, 2, such that [[phi].sub.1]([x.sub.1]) > 0, [[phi].sub.2]([x.sub.2]) > 0;

(b) [[integral].sub.[OMEGA]] [psi](x)[[phi].sub.1](x)dx = [alpha] > 0, [[integral].sub.[OMEGA]] [psi](x)[[phi].sub.2](x)dx = [beta] > 0.

Set

([phi] (t, x), u (t, x)) = [[integral].sub.[OMEGA]] [phi] (t, x) u (t, x) dx. (8)

Definition 1. The function u(t, x) is called a weak solution of initial-boundary value problem (1)-(3) if u(t, x) and [t.sup.b][[partial derivative].sub.t]u(t, x) are continuous with respect to t, if u satisfies (2) and (3), and if for all [phi] : (0, T) x [OMEGA] [right arrow] R which are twice continuously differentiable and satisfies

[mathematical expression not reproducible] (9)

Theorem 2. Assume that a weak solution of problem (1)-(2) with [mu] [greater than or equal to] [lambda] in (4) under assumptions (a)-(b) in the sense of Definition 1 exists; then there is some positive constant T such that

[mathematical expression not reproducible] (10)

[mathematical expression not reproducible] (11)

Proof. For [psi](x) defined in (6)-(7), set

F (t) = ([psi] (x), u (t, x)) = [[integral].sub.[OMEGA]] [psi] (x) u (t, x) dx, (12)

and then F(0) = [alpha] > 0 and

F' (t) = ([psi] (x), [[partial derivative].sub.t]u(t, x)) = [[integral].sub.[OMEGA]] [psi](x) [[partial derivative].sub.t]u(t, x)dx (13)

with [lim.sub.t[right arrow]0][t.sup.b]F'(t) = [beta] > 0. Since there is no differentiability of second order on the weak solution u with respect to t in Definition 1, then we will derive the second order of F(t) by use of (9). Multiplying t on (13), in terms of (8) and (9), we obtain

[mathematical expression not reproducible] (14)

According to the convexity of f(x) and Jensen's inequality, one has

([psi] (x), f (u (t, x))) [greater than or equal to] f (([psi] (x), u (t, x))) = f(F (t)). (15)

Then substitute (15) into (14), together with (4) for [lambda] = [mu]; we obtain

[mathematical expression not reproducible] (16)

which is equivalent to

d/dt ([t.sup.b]F' (t)) [greater than or equal to] [ct.sup.b-1][F.sup.1+p]. (17)

Note that F(0) = [alpha] > 0 and [lim.sub.t[right arrow]0][t.sup.b]F'(t) = [beta] > 0; then there exists a positive number [eta] such that F(t) > [alpha] for t [member of] (0, [eta]). Hence (17) means that [t.sup.b]F'(t) is strictly increasing on the interval (0, [eta]). Multiply [t.sup.b]F'(t) on (15) and then integrate it over ([epsilon], t) such that ([epsilon], t) [member of] (0, [epsilon]),

[([t.sup.b]F' (t)).sup.2] [greater than or equal to] ([[epsilon].sup.b]F' ([epsilon])).sup.2] + 2c [[integral].sup.t.sub.[epsilon]] [[tau].sup.2b-1] [F.sup.1+p] ([tau]) F' ([tau]) d[tau]. (18)

It follows that F'(t) > [[beta].sup.t-b] and F(t) > [alpha] on the entire existence interval (0, [eta]).

Now, for e >0, we set

H (t) = [[integral].sup.t.sub.[epsilon]] [[tau].sup.2b-1][F.sup.1+p] ([tau]) F' ([tau]) d[tau], (19)

G(t) = [(2 + p).sup.-1] [F.sup.2+p] (t). (20)

On one hand, it is easy to verify that H'(t) = [t.sup.2b-1]G'(t) holds and H(t) > 0, G(t) > 0. For b [greater than or equal to] 1/2, integrating by parts, it follows that

H (t) [less than or equal to] [t.sup.2b-1] G (t) - ([[epsilon].sup.2b-1] G ([epsilon]) + (2b - 1) [[integral].sup.t.sub.[epsilon]] [[tau].sup.2(b-1)] G ([tau]) d[tau]) [less than or equal to] [t.sup.2b-1] G (t). (21)

On the other hand, in terms of (17)-(20), we derive that

[(H' (t)).sup.2] = [t.sup.2b-2] [F.sup.2+2p] (t) [([t.sup.b] F'(t)).sup.2] [greater than or equal to] [(2 + p).sup.2+2p] x [t.sup.2(b-1)] [(([[epsilon].sup.b] F' ([epsilon])).sup.2] + 2cH (t)) [G.sup.1+p/(2+p)] (t). (22)

Then, substituting (21) into (22), we obtain

[(H'(t)).sup.2] [greater than or equal to] [(2 + p).sup.2+2p] x [t.sup.-1+p(1-2b)//(2+p)] [(([[epsilon].sup.b] F' ([epsilon])).sup.2] + 2ch (t)) x [H.sup.1+p/(2+p)] (t), (23)

which is equivalent to

[[(([[epsilon].sup.b] F' ([epsilon])).sup.2] + 2cH (t)) [H.sup.1+p/(2+p)] (t)].sup.-1/2 H' (t) [greater than or equal to] [(2 + p).sup.1+p] [t.sup.-1/2+p(1-2b)/2(2+p)]. (24)

For b [member of] (1/2, 1), integrate inequality (24) over ([epsilon], t); then

[mathematical expression not reproducible] (25)

Fix e; inequality (25) implies that there is a singularity of H(t) at some point T [less than or equal to] [T.sub.0], where

[mathematical expression not reproducible] (26)

Then, for some [t.sub.0] [less than or equal to] [T.sub.0], we derive

[mathematical expression not reproducible] (27)

Moreover, by a direct computation on (21), we have

[mathematical expression not reproducible] (28)

Hence, combining (20) through (27) and (28), we confirm that

[mathematical expression not reproducible] (29)

Finally, because of [psi](x) > 0, in terms of Holder's inequality, we have

[mathematical expression not reproducible] (30)

which implies (11). In particular, since [psi](x) satisfies (7), then, from (30), we obtain

[mathematical expression not reproducible] (31)

This derives (10).

Next, we consider the case for b [member of] [0, 1/2). Therefore 2b - 1 < 0. Then, for 0 < [epsilon] [less than or equal to] [tau] [less than or equal to] t, we have [t.sup.2b-1] [less than or equal to] [[tau].sup.2b-1] [less than or equal to] [[epsilon].sup.2b-1]. Since c > 0, p > 0, F(t) > 0, and F'(t) > 0 on the entire existence interval from (18), it follows that

[([t.sup.b] F' (t)).sup.2] [greater than or equal to] [t.sup.2b-1] [[epsilon] [(F' ([epsilon])).sup.2] + 2c [[integral].sup.t.sub.[epsilon]] [F.sup.1+p] ([tau]) F' ([tau]) d[tau]]. (32)

This is equivalent to

[(F' (t)).sup.2] [greater than or equal to] [t.sup.-1] [[epsilon] [(F' ([epsilon])).sup.2] + 2c/2 + p ([F.sup.2+p] (t) - [F.sup.2+p] ([epsilon])). (33)

Note F(t) is increasing; then

[[[epsilon] [(F' ([epsilon])).sup.2] + 2c/2 + p ([F.sup.2+p] (t) - [F.sup.2+p] ([epsilon])).sup.-1/2 F' (t) [greater than or equal to] [t.sup.-1/2. (34)

Fix [epsilon] > 0 and for t > [epsilon] and integrate (34) on [[epsilon], t]; it follows that

[[integral].sup.F(t).sub.F([epsilon])] [[epsilon] [[(F' ([epsilon])).sup.2] 2c/2 + p ([y.sup.2+p] - [F.sup.2+p] ([epsilon]))].sup.-1/2 dy [greater than or equal to] 2 ([t.sup.1/2] - [[epsilon].sup.1/2]) > 0. (35)

Set [T.sub.1] = [[integral].sup.+[infinity].sub.[alpha]] [[[epsilon] [(F'([epsilon])).sup.2] + (2c/(2 + p))([y.sup.2+p] - [F.sup.2+p]([epsilon]))].sup.-1/2] dy; then (35) implies that there exists some point T < [T.sub.1] such that

[mathematical expression not reproducible] (36)

Hence, there is T [member of] (0, [T.sub.1]) such that (30) holds for b [member of] (0, 1/2). Then (10) and (11) are yielded by use of Holder'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 ideaofproofis similar to that used in Section 2, but it is more complicated in calculus.

Let [psi](x) denote the first eigenfunction of Dirichlet problem (6); then there is only one eigenfunction [psi](x) > 0 satisfying (7). As shown in [24], set [S.sub.r] = [x : [absolute value of (x)] < r] = [OMEGA] in (6) and denote [[psi].sub.r](x) as the eigenfunction corresponding to the first eigenvalue [[lambda].sub.r]; then [[psi].sub.r](x) is also the function of [absolute value of (x)] only and [[lambda].sub.r] decrease with r.

Assume that

(c) nonnegative functions [[phi].sub.1](x), [[phi].sub.2](x) have support in [S.sub.R-[delta]] for some positive constant R and [delta] [member of] (0, R); there exists [x.sub.i] [member of] [S.sub.R-[delta]], i = 1, 2 such that [[phi].sub.1]([x.sub.1]) > 0 and [[phi].sub.2]([x.sub.2]) > 0;

(d) [mathematical expression not reproducible]

Set

[mathematical expression not reproducible] (37)

Taking a similar procedure of the proof of (36) and (46) in [25] with the principle of superposition, we can obtain a solution of problem (1)-(2) which is the representation at a point (t, x) 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 (t, x). 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 R) whenever [[phi].sub.i](x), i = 1, 2 has support in [x [member of] [R.sup. ] | [absolute value of (x)] < R], u(t, x) has support in [x [member of] [R.sup.n] | [absolute value of (x)] < R + y(t)] for all t [member of] (0, T), where

[gamma](t) = 2[square root of (t)]. (38)

Now fix r > R and the value will be given later. Let t [less than or equal to] [[gamma].sup.-1](r - R); then u(t, x) given in Definition 3 has support in [S.sub.R-[delta]+[gamma](t)] x [t [less than or equal to] [[gamma].sup.-1] (r - R)], which is contained in [S.sub.r- [delta]] x [t [less than or equal to] [[gamma].sup.-1](r - R)].

Definition 4. The function u(t, x) is called a weak solution of problem (1)-(2) which enjoys the cone compact support property, if u(t, x) and [t.sup.b][[partial derivative].sub.t]u(t, x) are continuous with respect to t, if u satisfies (2), and if for all [phi] : [S.sub.R-[delta]] x (0, T) [right arrow] R which are twice continuously differentiable and satisfies

[mathematical expression not reproducible] (39)

Theorem 5. Assume that a weak solution of problem (1)-(2) with [mu] [greater than or equal to] [lambda] in (4) under conditions (c)-(d) in the sense of Definition 4 exists; then there is some positive constant T such that

[mathematical expression not reproducible] (40)

[mathematical expression not reproducible] (41)

Proof. Define a function [[PSI].sub.r](x); that is,

[mathematical expression not reproducible] (42)

where [[psi].sub.r] is first eigenfunction of Dirichlet problem (6); [[psi].sub.0] is a twice continuously differentiable function of [rho] = [absolute value of (x)] which satisfies [[psi].sup.i.sub.0] (r - [delta]/2) = [[psi].sup.i.sub.r](r - [delta]/2) and [[psi].sup.i.sub.0] (r) = 0 for i = 0, 1, 2. When t < [[gamma].sup.-1] (r-R), substitute [phi](t, x) by [[PSI].sub.r](t, x) in (39); then it is easy to verify that

t ([[psi].sub.r] (x), [[partial derivative].sub.t]u (t, x)) = [[integral].sup.t.sub.0] [(1 - b) ([[psi].sub.r] (x), [[partial derivative].sub.[tau]]u ([tau], x)) + ([DELTA][[psi].sub.r] (x), u([tau], [xi])) + ([[psi].sub.r] (x), f (u([tau], x)))] d[tau]. (43)

since the support of u(t, x) is contained in [S.sub.r-[delta]] x [0, [[gamma].sup.-1] (r-R)).

For t [less than or equal to] [[gamma].sup.-1](r - R), set

[mathematical expression not reproducible] (44)

and then, by the assumption,

[mathematical expression not reproducible] (45)

[mathematical expression not reproducible] (46)

In terms of (43) and (46), it follows that

[tF'.sub.r] (t) = [[integral].sup.t.sub.0] [(1 - b)([[psi].sub.r] (x), [[partial derivative].sub.[tau]]u ([tau], x)) + ([DELTA][[psi].sub.r] (x), u ([tau], x)) + ([[psi].sub.r] (x), f(u([tau], x)))]d[tau]. (47)

Therefore, we obtain

[F".sub.r] (t) = ([t.sup.-1] ([tF'.sub.r] (t)))' = [t.sup.-1] [-b ([[psi].sub.r] (x), [[partial derivative].sub.t]u (t, x)) + ([[psi].sub.r] (x), f (u (t, x))) - [[lambda].sub.r] ([[psi].sub.r] (x), u (t, x))], (48)

which is equivalent to

d/dt ([t.sup.b]F' (t)) [greater than or equal to] [ct.sup.b-1] (([[psi].sub.r] (x), f (u (t, x))) - [[lambda].sub.r] ([[psi].sub.r] (x), u(t, x))). (49)

In terms of the convexity of f(x) and Jensen's inequality, one has

[mathematical expression not reproducible] (50)

Then substituting (50) into (49), together with (4) for [[lambda].sub.r] = [mu], we obtain

d/dt ([t.sup.b]F' (t)) [greater than or equal to] [ct.sup.b-1] [F.sup.1+p.sub.r] (t). (51)

In fact, if set f(s) = [s.sup.1+p], then we can choose R such that [[lambda].sub.r] < [[alpha].sup.p] for Dirichlet problem (6) in [S.sub.r], and then (51) holds which is independent of (4).

Note that [F.sub.r](0) = [alpha] > 0 and [lim.sub.t[right arrow]0][t.sup.b][F'.sub.r](0) = [beta] > 0; then there exists an interval (0, [eta]) such that [F.sub.r](t) > [alpha] for t [member of] (0, [eta]). Hence, on (0, [eta]),

d/dt([t.sup.b][F'.sub.r] (t)) [greater than or equal to] [ct.sup.b-1][F.sup.1+p.sub.t] (t) > 0. (52)

This means that [t.sup.b][F'.sub.r](t) is strictly increasing on (0, [eta]). Then we obtain [F'.sub.r](t) > [beta][t.sup.-b] > 0 on (0, [eta]). Multiply [t.sup.b][F'.sub.r](t) on (51) and then integrate it over ([epsilon], t) where ([epsilon], t) [member of] (0, [eta]):

[([t.sup.b][F'.sub.r] (t)).sup.2] [greater than or equal to] [([[epsilon].sup.b][F'.sub.r] ([epsilon])).sup.2] + 2c [[integral].sup.t.sub.[epsilon]] [[tau].sup.2b-1][F.sup.1+p.sub.r] ([tau]) [F'.sub.r] ([tau]) d[tau]. (53)

It follows that [F'.sub.r](t) > [beta][t.sup.-b] and [F.sub.r](t) > [alpha] on the entire existence interval.

Now, for [epsilon] > 0, we set

[H.sub.r] (t) = [[integral].sup.t.sub.[epsilon]] [[tau].sup.2b-1] [F.sup.1+p.sub.r] ([tau]) [F'.sub.r] ([tau]) d[tau], (54)

[G.sub.r] (t) = [(2 + p).sup.-1] [F.sup.2+p.sub.r] (t). (55)

Then, there, [H'.sub.r](t) = [t.sup.2b-1][G'.sub.r](t) holds and [H.sub.r](t) > 0, [G.sub.r](t) > 0. For b [member of] [1/2, 1), integrating by parts, it follows that

[mathematical expression not reproducible] (56)

Combining to (51)-(56), we derive that

[([H'.sub.r] (t)).sup.2] [greater than or equal to] [(2 + P).sup.2+2p] x [t.sup.-1+p(1-2b)/(2+p)] ([([[epsilon].sup.b][F'.sub.r] ([epsilon])).sup.2] + 2c[H.sub.r] (t)) x [H.sup.1+p/(2+p).sub.r] (t), (57)

which is equivalent to

[[([([[epsilon].sup.b][F'.sub.r] ([epsilon])).sup.2] + 2cH (t)) [H.sup.1+p/(2+p).sub.r] (t)].sup.-1/2] [H'.sub.r] (t) [greater than or equal to] [(2 + p).sup.1+p] [t.sup.-1/2+p(1-2b)/2(2+p)]. (58)

Then, for b [member of] [1/2, 1), integrating (58) over ([epsilon], t), we obtain

[mathematical expression not reproducible] (59)

According to the previous analysis, we have [[epsilon].sup.b][F'.sub.r]([epsilon]) > [beta] > 0 independent of r; then

[mathematical expression not reproducible] (60)

Set [K.sub.0] = [[integral].sup.+[infinity].sub.[beta]] [(2ct).sup.-1/2][t.sup.-1/2-p/2(2+p)]dt, in terms of (59)-(60); then [K.sub.0] is a finite positive number independent of r and [epsilon], and

[(2 + p).sup.1+p] ([t.sup.1/2+p(1-2b)/2(2+p)] - [[epsilon].sup.1/2+p(1-2b)/2(2+p)]) < [K.sub.0]. (61)

Now fix [epsilon]; we choose r = [r.sub.0],

[r.sub.0] = [([K.sub.0] (2 + p).sup.-1-p] + [[epsilon].sup.1/2+p(1-2b)/2(2+p)]).sup.1/(1/2+p(1-2b)/2(2+p))] + 2R, (62)

and then we obtain

t < [r.sub.0] - 2R < [r.sub.0] - R (63)

for all t [member of] ([epsilon], [eta]) [intersection] (0, [r.sub.0] - R), where (0, [eta]) is the existence interval of solution (1)-(2). Hence there is a T [less than or equal to] [r.sub.0] - 2R, such that

[mathematical expression not reproducible] (64)

From (55)-(56) and (64), we obtain

[mathematical expression not reproducible] (65)

At last, by use of Holder inequality on (65), we derive (40) and (41) for b [member of] [1/2, 1).

For b [member of] (0, 1/2), we also obtain (35) with the term F being substituted by [F.sub.r] by a procedure as mentioned in Section 2; that is,

[mathematical expression not reproducible] (66)

Since we do not know the sign of the term [epsilon][([F'.sub.r]([epsilon])).sup.2] - (2c/(2 + p))[F.sup.2+p.sub.r]([epsilon]), we calculate

[mathematical expression not reproducible] (67)

where we use p > 0 and [F.sub.r]([epsilon]) > [alpha].

Let [K.sub.1] = [(2c/(2+p)).sup.-1/2][[alpha].sup.-p/2] [[integral].sup.+[infinity].sub.1][([z.sup.2+p]-1).sup.-1/2] dz which is a finite positive constant independent of r and [epsilon]; then

[K.sub.1] > 2 ([t.sup.1/2] - [[epsilon].sup.1/2]). (68)

Fixing [epsilon], we choose r = [r.sub.1]:

[r.sub.1] = [([[epsilon].sup.1/2] + 1/2 [K.sub.1]).sup.2] + 2R, (69)

and then it follows from (68) that

t < [r.sub.1] - 2R < [r.sub.1] - R (70)

for all t [member of] ([epsilon], q) [intersection] (0, [r.sub.1] - R), where (0, [eta]) is the existence interval of solution (1)-(2). Hence there is T [less than or equal to] [r.sub.1] - 2R, such that

[mathematical expression not reproducible] (71)

By a computation with (55)-(56) and (71), we obtain

[mathematical expression not reproducible] (72)

Hence, in terms of Holder inequality, the last formula implies (40) and (41) for b [member of] (0, 1/2).

Then we conclude that Theorem 5 holds.

https://doi.org/10.1155/2018/3931297

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).

References

[1] T. Hristov, "Singular solutions to Protter problem for Keldysh type equations," in Proceedings of the 40th International Conference of Applications of Mathematics in Engineering and Economics, AMEE 2014, pp. 255-262, Bulgaria, June 2014.

[2] T. H. Otway, The Dirichlet Problem for Elliptic-Hyperbolic Equations of Keldysh Type, vol. 2043 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2012.

[3] S. Chen, "Mixed type equations in gas dynamics," Quarterly of Applied Mathematics, vol. 68, no. 3, pp. 487-511, 2010.

[4] C. S. Morawetz, "Mixed equations and transonic flow," Journal of Hyperbolic Differential Equations, vol. 1, no. 1, pp. 1-26, 2004.

[5] S. Chen, "The fundamental solution of the Keldysh type operator," Science China Mathematics, vol. 52, no. 9, pp. 1829-1843, 2009.

[6] Nedyu Popivanov, Tsvetan Hristov, Aleksey Nikolov, and Manfred Schneider, "Singular Solutions to a (3 + 1)-D Protter-Morawetz Problem for Keldysh-Type Equations," Advances in Mathematical Physics, vol. 2017, Article ID 1571959, 16 pages, 2017.

[7] N. Popivanov, T. Hristov, A. Nikolov, and M. Schneider, "On the existence and uniqueness of a generalized solution of the Protter problem for (3 + 1)-D Keldysh-type equations," Boundary Value Problems, vol. 2017, no. 26, 2017.

[8] P. Brenner and W. von Wahl, "Global classical solutions of nonlinear wave equations," Mathematische Zeitschrift, vol. 176, no. 1, pp. 87-121, 1981.

[9] M. G. Grillakis, "Regularity and asymptotic behaviour of the wave equation with a critical nonlinearity," Annals of Mathematics: Second Series, vol. 132, no. 3, pp. 485-509, 1990.

[10] V. Georgiev, H. Lindblad, and C. . Sogge, "Weighted Strichartz estimates and global existence for semilinear wave equations," American Journal of Mathematics, vol. 119, no. 6, pp. 1291-1319, 1997.

[11] K. Jorgens, "Das Anfangswertproblem in Grossen fur eine klasse nichtlinearer Wellengleichungen," Mathematische Zeitschrift, vol. 77, no. 1, pp. 295-308, 1961.

[12] J. B. Keller, "On solutions of nonlinear wave equations," Communications on Pure and Applied Mathematics, vol. 10, pp. 523530, 1957.

[13] K. Jorgens, "Nonlinear wave equations," in Nonlinear Wave Equations, vol. 2, University of Colorado, Department of Mathematics, Denver, Colo, USA, 2017.

[14] R. T. Glassey, "Blow-up theorems for nonlinear wave equations," Mathematische Zeitschrift, vol. 132, pp. 183-203, 1973.

[15] X. Liu and Y. Zhou, "Global nonexistence of solutions to a semilinear wave equation in the Minkowski space," Applied Mathematics Letters, vol. 21, no. 8, pp. 849-854, 2008.

[16] S. A. Messaoudi, "Blow up in a nonlinearly damped wave equation," Mathematische Nachrichten, vol. 231, no. 1, pp. 1-7, 2001.

[17] T. C. Sideris, "Nonexistence of global solutions to semilinear wave equations in high dimensions," Journal of Differential Equations, vol. 52, no. 3, pp. 378-406, 1984.

[18] B. T. Yordanov and Q. S. Zhang, "Finite time blow up for critical wave equations in high dimensions," Journal of Functional Analysis, vol. 231, no. 2, pp. 361-374, 2006.

[19] S. Kaplan, "On the growth of solutions of quasi-linear parabolic equations," Communications on Pure and Applied Mathematics, vol. 16, pp. 305-330, 1963.

[20] H. Fujita, "On the blowing up of solutions of the Cauchy problem for [partial derivative]tu=[DELTA]u+u1+[alpha][J]," Journal of the Faculty of Science, vol. 13, 1966.

[21] H. Fujita, "On some nonexistence and nonuniqueness theorems for nonlinear parabolic equations," Proceedings of Symposia in Pure Mathematics, vol. 18, pp. 105-113, 1970.

[22] H. A. Levine, "On the nonexistence of global solutions to a nonlinear Euler-Poisson-Darboux equation," Journal of Mathematical Analysis and Applications, vol. 48, pp. 646-651, 1974.

[23] H. A. Levine, "Growth of solutions of generalized nonlinear Euler-Poisson-Darboux equations," Archive for Rational Mechanics and Analysis, vol. 61, no. 1, pp. 77-89, 1976.

[24] R. Courant and D. Hilbert, Methods of Mathematical Physics, vol. 1, CUP Archive, 1966.

[25] K. Zhang, "The Cauchy problem for semilinear hyperbolic equation with characteristic degeneration on the initial hyperplane," Mathematical Methods in the Applied Sciences, vol. 41, no. 6, pp. 2429-2441, 2018.

[26] A. Erdlyi, W. Magnus, F. Oberhettinger et al., Higher Transcendental Functions, vol. 1, McGraw-Hill, New York, NY, USA, 1953.

Kangqun Zhang (iD)

Department of Mathematics and Physics, Nanjing Institute of Technology, Nanjing 211167, China

Correspondence should be addressed to Kangqun Zhang; chkqnju@hotmail.com

Received 28 April 2018; Revised 17 June 2018; Accepted 16 July 2018; Published 1 August 2018

Academic Editor: Ming Mei
COPYRIGHT 2018 Hindawi Limited
No portion of this article can be reproduced without the express written permission from the copyright holder.
Copyright 2018 Gale, Cengage Learning. All rights reserved.

Article Details
Printer friendly Cite/link Email Feedback
Title Annotation:Research Article
Author:Zhang, Kangqun
Publication:Advances in Mathematical Physics
Article Type:Report
Geographic Code:1USA
Date:Jan 1, 2018
Words:4691
Previous Article:The Quantum Cheshire Cat Effect in the Presence of Decoherence.
Next Article:Motion of a Spot in a Reaction Diffusion System under the Influence of Chemotaxis.
Topics:

Terms of use | Privacy policy | Copyright © 2021 Farlex, Inc. | Feedback | For webmasters